2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.4%

Time: 19.8s

Alternatives: 13

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: 62.3% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := t\_2 + 1\\ \varepsilon \cdot {\left({\left(\left(t\_2 + \varepsilon \cdot \left(\frac{\sin x \cdot t\_3}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \left(\frac{t\_0 \cdot \left(-1 - t\_2\right)}{t\_1} + \left(t\_3 \cdot -0.5 + 0.16666666666666666 \cdot t\_2\right)\right)\right)\right)\right) + 1\right)}^{3}\right)}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0)))
   (*
    eps
    (pow
     (pow
      (+
       (+
        t_2
        (*
         eps
         (-
          (/ (* (sin x) t_3) (cos x))
          (*
           eps
           (+
            0.16666666666666666
            (+
             (/ (* t_0 (- -1.0 t_2)) t_1)
             (+ (* t_3 -0.5) (* 0.16666666666666666 t_2))))))))
       1.0)
      3.0)
     0.3333333333333333))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * pow(pow(((t_2 + (eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2)))))))) + 1.0), 3.0), 0.3333333333333333);
}

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
    real(8) :: t_2
    real(8) :: t_3
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = t_2 + 1.0d0
    code = eps * ((((t_2 + (eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666d0 + (((t_0 * ((-1.0d0) - t_2)) / t_1) + ((t_3 * (-0.5d0)) + (0.16666666666666666d0 * t_2)))))))) + 1.0d0) ** 3.0d0) ** 0.3333333333333333d0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = Math.pow(Math.cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * Math.pow(Math.pow(((t_2 + (eps * (((Math.sin(x) * t_3) / Math.cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2)))))))) + 1.0), 3.0), 0.3333333333333333);
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = t_0 / t_1
	t_3 = t_2 + 1.0
	return eps * math.pow(math.pow(((t_2 + (eps * (((math.sin(x) * t_3) / math.cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2)))))))) + 1.0), 3.0), 0.3333333333333333)

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	return Float64(eps * ((Float64(Float64(t_2 + Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) - Float64(eps * Float64(0.16666666666666666 + Float64(Float64(Float64(t_0 * Float64(-1.0 - t_2)) / t_1) + Float64(Float64(t_3 * -0.5) + Float64(0.16666666666666666 * t_2)))))))) + 1.0) ^ 3.0) ^ 0.3333333333333333))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = t_2 + 1.0;
	tmp = eps * ((((t_2 + (eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2)))))))) + 1.0) ^ 3.0) ^ 0.3333333333333333);
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, N[(eps * N[Power[N[Power[N[(N[(t$95$2 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.16666666666666666 + N[(N[(N[(t$95$0 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(t$95$3 * -0.5), $MachinePrecision] + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.6%

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

5. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \color{blue}{{\left({\left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-1 \cdot \varepsilon, 0.16666666666666666 + \mathsf{fma}\left(-1, {\sin x}^{2} \cdot \frac{1 + {\sin x}^{2} \cdot {\cos x}^{-2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.16666666666666666, {\sin x}^{2} \cdot {\cos x}^{-2}, \left(1 + {\sin x}^{2} \cdot {\cos x}^{-2}\right) \cdot -0.5\right)\right), \sin x \cdot \frac{1 + {\sin x}^{2} \cdot {\cos x}^{-2}}{\cos x}\right), {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in eps around 0 99.6%

   \[\leadsto \varepsilon \cdot {\left({\left(1 + \color{blue}{\left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

7. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot {\left({\left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \left(\frac{{\sin x}^{2} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \cdot -0.5 + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right) + 1\right)}^{3}\right)}^{0.3333333333333333} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := t\_2 + 1\\ \varepsilon \cdot {\left({\left(\left(\varepsilon \cdot \left(\frac{\sin x \cdot t\_3}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \left(\frac{t\_0 \cdot \left(-1 - t\_2\right)}{t\_1} + \left(t\_3 \cdot -0.5 + 0.16666666666666666 \cdot t\_2\right)\right)\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t\_1}\right) + 1\right)}^{3}\right)}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0)))
   (*
    eps
    (pow
     (pow
      (+
       (+
        (*
         eps
         (-
          (/ (* (sin x) t_3) (cos x))
          (*
           eps
           (+
            0.16666666666666666
            (+
             (/ (* t_0 (- -1.0 t_2)) t_1)
             (+ (* t_3 -0.5) (* 0.16666666666666666 t_2)))))))
        (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_1))
       1.0)
      3.0)
     0.3333333333333333))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * pow(pow((((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_1)) + 1.0), 3.0), 0.3333333333333333);
}

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
    real(8) :: t_2
    real(8) :: t_3
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = t_2 + 1.0d0
    code = eps * (((((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666d0 + (((t_0 * ((-1.0d0) - t_2)) / t_1) + ((t_3 * (-0.5d0)) + (0.16666666666666666d0 * t_2))))))) + ((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / t_1)) + 1.0d0) ** 3.0d0) ** 0.3333333333333333d0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = Math.pow(Math.cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * Math.pow(Math.pow((((eps * (((Math.sin(x) * t_3) / Math.cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + ((0.5 - (Math.cos((x * 2.0)) / 2.0)) / t_1)) + 1.0), 3.0), 0.3333333333333333);
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = t_0 / t_1
	t_3 = t_2 + 1.0
	return eps * math.pow(math.pow((((eps * (((math.sin(x) * t_3) / math.cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + ((0.5 - (math.cos((x * 2.0)) / 2.0)) / t_1)) + 1.0), 3.0), 0.3333333333333333)

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	return Float64(eps * ((Float64(Float64(Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) - Float64(eps * Float64(0.16666666666666666 + Float64(Float64(Float64(t_0 * Float64(-1.0 - t_2)) / t_1) + Float64(Float64(t_3 * -0.5) + Float64(0.16666666666666666 * t_2))))))) + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_1)) + 1.0) ^ 3.0) ^ 0.3333333333333333))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = t_2 + 1.0;
	tmp = eps * (((((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_1)) + 1.0) ^ 3.0) ^ 0.3333333333333333);
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, N[(eps * N[Power[N[Power[N[(N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.16666666666666666 + N[(N[(N[(t$95$0 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(t$95$3 * -0.5), $MachinePrecision] + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.6%

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

5. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \color{blue}{{\left({\left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-1 \cdot \varepsilon, 0.16666666666666666 + \mathsf{fma}\left(-1, {\sin x}^{2} \cdot \frac{1 + {\sin x}^{2} \cdot {\cos x}^{-2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.16666666666666666, {\sin x}^{2} \cdot {\cos x}^{-2}, \left(1 + {\sin x}^{2} \cdot {\cos x}^{-2}\right) \cdot -0.5\right)\right), \sin x \cdot \frac{1 + {\sin x}^{2} \cdot {\cos x}^{-2}}{\cos x}\right), {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in eps around 0 99.6%

   \[\leadsto \varepsilon \cdot {\left({\left(1 + \color{blue}{\left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
7. Step-by-step derivation

   1. unpow2 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   2. sin-mult 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

8. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]
9. Step-by-step derivation

   1. div-sub 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   2. +-inverses 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   3. cos-0 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   4. metadata-eval 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   5. count-2 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   6. *-commutative 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

10. Simplified 99.6%

    \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

11. Final simplification 99.6%

    \[\leadsto \varepsilon \cdot {\left({\left(\left(\varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \left(\frac{{\sin x}^{2} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \cdot -0.5 + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) + 1\right)}^{3}\right)}^{0.3333333333333333} \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := t\_2 + 1\\ \varepsilon \cdot \left(t\_2 + \left(\varepsilon \cdot \left(\frac{\sin x \cdot t\_3}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \left(\frac{t\_0 \cdot \left(-1 - t\_2\right)}{t\_1} + \left(t\_3 \cdot -0.5 + 0.16666666666666666 \cdot t\_2\right)\right)\right)\right) + 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0)))
   (*
    eps
    (+
     t_2
     (+
      (*
       eps
       (-
        (/ (* (sin x) t_3) (cos x))
        (*
         eps
         (+
          0.16666666666666666
          (+
           (/ (* t_0 (- -1.0 t_2)) t_1)
           (+ (* t_3 -0.5) (* 0.16666666666666666 t_2)))))))
      1.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * (t_2 + ((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + 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
    real(8) :: t_2
    real(8) :: t_3
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = t_2 + 1.0d0
    code = eps * (t_2 + ((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666d0 + (((t_0 * ((-1.0d0) - t_2)) / t_1) + ((t_3 * (-0.5d0)) + (0.16666666666666666d0 * t_2))))))) + 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 = Math.pow(Math.cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	return eps * (t_2 + ((eps * (((Math.sin(x) * t_3) / Math.cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + 1.0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = t_0 / t_1
	t_3 = t_2 + 1.0
	return eps * (t_2 + ((eps * (((math.sin(x) * t_3) / math.cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + 1.0))

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	return Float64(eps * Float64(t_2 + Float64(Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) - Float64(eps * Float64(0.16666666666666666 + Float64(Float64(Float64(t_0 * Float64(-1.0 - t_2)) / t_1) + Float64(Float64(t_3 * -0.5) + Float64(0.16666666666666666 * t_2))))))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = t_2 + 1.0;
	tmp = eps * (t_2 + ((eps * (((sin(x) * t_3) / cos(x)) - (eps * (0.16666666666666666 + (((t_0 * (-1.0 - t_2)) / t_1) + ((t_3 * -0.5) + (0.16666666666666666 * t_2))))))) + 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[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, N[(eps * N[(t$95$2 + N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.16666666666666666 + N[(N[(N[(t$95$0 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(t$95$3 * -0.5), $MachinePrecision] + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} - \varepsilon \cdot \left(0.16666666666666666 + \left(\frac{{\sin x}^{2} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \cdot -0.5 + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + 1\right)\right) \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.6%

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

5. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \color{blue}{{\left({\left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-1 \cdot \varepsilon, 0.16666666666666666 + \mathsf{fma}\left(-1, {\sin x}^{2} \cdot \frac{1 + {\sin x}^{2} \cdot {\cos x}^{-2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.16666666666666666, {\sin x}^{2} \cdot {\cos x}^{-2}, \left(1 + {\sin x}^{2} \cdot {\cos x}^{-2}\right) \cdot -0.5\right)\right), \sin x \cdot \frac{1 + {\sin x}^{2} \cdot {\cos x}^{-2}}{\cos x}\right), {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in eps around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. associate-/l* 99.5%

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

   3. associate-/l* 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 98.8% 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 62.9%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 99.3%

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

5. Final simplification 99.3%

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

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

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(Float64(eps * (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[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

   1. pow-pow 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1 99.1%

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

   4. distribute-rgt-in 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. div-inv 99.1%

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

   7. pow-flip 99.1%

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

   8. metadata-eval 99.1%

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in x around inf 99.1%

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

Alternative 7: 98.7% 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 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

   1. pow-pow 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1 99.1%

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

   4. distribute-rgt-in 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. div-inv 99.1%

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

   7. pow-flip 99.1%

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

   8. metadata-eval 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 8: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

   1. pow-pow 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1 99.1%

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

   4. distribute-rgt-in 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. div-inv 99.1%

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

   7. pow-flip 99.1%

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

   8. metadata-eval 99.1%

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

9. Applied egg-rr 99.1%

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

    1. unpow2 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    2. sin-mult 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

11. Applied egg-rr 99.1%

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

    1. div-sub 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    2. +-inverses 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    3. cos-0 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    4. metadata-eval 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    5. count-2 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    6. *-commutative 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

13. Simplified 99.1%

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

14. Final simplification 99.1%

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

Alternative 9: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. *-commutative 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 10: 98.1% 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 62.9%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 98.6%

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

   1. unpow2 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 11: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 12: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

   1. unpow2 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   2. sin-mult 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

9. Applied egg-rr 35.5%

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

    1. div-sub 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    2. +-inverses 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    3. cos-0 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    4. metadata-eval 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    5. count-2 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    6. *-commutative 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

11. Simplified 35.5%

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

12. Taylor expanded in x around 0 98.5%

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

    1. *-rgt-identity 98.5%

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

    2. distribute-lft-out 98.5%

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

    3. +-commutative 98.5%

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

    4. unpow2 98.5%

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

    5. fma-define 98.5%

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

14. Simplified 98.5%

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

Alternative 13: 97.6% 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 62.9%

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

   1. add-cbrt-cube 25.5%

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

   2. pow1/3 24.9%

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

   3. pow3 24.9%

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

4. Applied egg-rr 24.9%

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

5. Taylor expanded in eps around 0 35.5%

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

   1. sub-neg 35.5%

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

   2. mul-1-neg 35.5%

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

   3. remove-double-neg 35.5%

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

7. Simplified 35.5%

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

   1. unpow2 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   2. sin-mult 99.6%

      \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

9. Applied egg-rr 35.5%

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

    1. div-sub 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    2. +-inverses 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    3. cos-0 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    4. metadata-eval 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    5. count-2 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

    6. *-commutative 99.6%

       \[\leadsto \varepsilon \cdot {\left({\left(1 + \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

11. Simplified 35.5%

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

12. Taylor expanded in x around 0 98.2%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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