2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.6%

Time: 29.4s

Alternatives: 10

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := -1 - t\_0\\ t_3 := 1 + t\_0\\ t_4 := 0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, t\_3, t\_0 \cdot 0.16666666666666666\right) + t\_0 \cdot t\_2\right)\\ \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{t\_3}}\right) + \left({\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(t\_1 \cdot t\_2\right) - t\_4 \cdot t\_1\right) - {\varepsilon}^{3} \cdot t\_4\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_1 (/ (sin x) (cos x)))
        (t_2 (- -1.0 t_0))
        (t_3 (+ 1.0 t_0))
        (t_4
         (+
          0.16666666666666666
          (+ (fma -0.5 t_3 (* t_0 0.16666666666666666)) (* t_0 t_2)))))
   (+
    (fma
     eps
     (+ 1.0 (pow (tan x) 2.0))
     (/ (pow eps 2.0) (/ (/ (cos x) (sin x)) t_3)))
    (-
     (* (pow eps 4.0) (- (* -0.3333333333333333 (* t_1 t_2)) (* t_4 t_1)))
     (* (pow eps 3.0) t_4)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_1 = sin(x) / cos(x);
	double t_2 = -1.0 - t_0;
	double t_3 = 1.0 + t_0;
	double t_4 = 0.16666666666666666 + (fma(-0.5, t_3, (t_0 * 0.16666666666666666)) + (t_0 * t_2));
	return fma(eps, (1.0 + pow(tan(x), 2.0)), (pow(eps, 2.0) / ((cos(x) / sin(x)) / t_3))) + ((pow(eps, 4.0) * ((-0.3333333333333333 * (t_1 * t_2)) - (t_4 * t_1))) - (pow(eps, 3.0) * t_4));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_1 = Float64(sin(x) / cos(x))
	t_2 = Float64(-1.0 - t_0)
	t_3 = Float64(1.0 + t_0)
	t_4 = Float64(0.16666666666666666 + Float64(fma(-0.5, t_3, Float64(t_0 * 0.16666666666666666)) + Float64(t_0 * t_2)))
	return Float64(fma(eps, Float64(1.0 + (tan(x) ^ 2.0)), Float64((eps ^ 2.0) / Float64(Float64(cos(x) / sin(x)) / t_3))) + Float64(Float64((eps ^ 4.0) * Float64(Float64(-0.3333333333333333 * Float64(t_1 * t_2)) - Float64(t_4 * t_1))) - Float64((eps ^ 3.0) * t_4)))
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]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(0.16666666666666666 + N[(N[(-0.5 * t$95$3 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   2. add-sqr-sqrt 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \cdot \sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   3. pow2 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   4. sqrt-div 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\left(\frac{\sqrt{{\sin x}^{2}}}{\sqrt{{\cos x}^{2}}}\right)}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   5. unpow2 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sqrt{\color{blue}{\sin x \cdot \sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   6. sqrt-prod 53.2%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   7. add-sqr-sqrt 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sin x}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   8. unpow2 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\sqrt{\color{blue}{\cos x \cdot \cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   9. sqrt-prod 98.9%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   10. add-sqr-sqrt 99.7%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\cos x}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   11. tan-quot 99.7%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\tan x}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot {\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
8. Step-by-step derivation

   1. *-lft-identity 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

10. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) + \left({\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \frac{\sin x}{\cos x}\right) - {\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 2: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ t_2 := t\_1 - t\_0 \cdot -0.3333333333333333\\ t_3 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_4 := \frac{\sin x \cdot t\_2}{\cos x}\\ \left(\varepsilon \cdot \left(1 + t\_3\right) + \left({\varepsilon}^{2} \cdot \left(t\_0 + t\_1\right) + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t\_3 + t\_4\right)\right) + {\varepsilon}^{4} \cdot \left(t\_2 + \left(\frac{\sin x \cdot \left(t\_4 - t\_3 \cdot -0.3333333333333333\right)}{\cos x} + t\_0 \cdot 0.3333333333333333\right)\right)\right)\right)\right) + \mathsf{fma}\left(-1, \tan x, \tan x\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (t_2 (- t_1 (* t_0 -0.3333333333333333)))
        (t_3 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_4 (/ (* (sin x) t_2) (cos x))))
   (+
    (+
     (* eps (+ 1.0 t_3))
     (+
      (* (pow eps 2.0) (+ t_0 t_1))
      (+
       (* (pow eps 3.0) (+ 0.3333333333333333 (+ t_3 t_4)))
       (*
        (pow eps 4.0)
        (+
         t_2
         (+
          (/ (* (sin x) (- t_4 (* t_3 -0.3333333333333333))) (cos x))
          (* t_0 0.3333333333333333)))))))
    (fma -1.0 (tan x) (tan x)))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = pow(sin(x), 3.0) / pow(cos(x), 3.0);
	double t_2 = t_1 - (t_0 * -0.3333333333333333);
	double t_3 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_4 = (sin(x) * t_2) / cos(x);
	return ((eps * (1.0 + t_3)) + ((pow(eps, 2.0) * (t_0 + t_1)) + ((pow(eps, 3.0) * (0.3333333333333333 + (t_3 + t_4))) + (pow(eps, 4.0) * (t_2 + (((sin(x) * (t_4 - (t_3 * -0.3333333333333333))) / cos(x)) + (t_0 * 0.3333333333333333))))))) + fma(-1.0, tan(x), tan(x));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))
	t_2 = Float64(t_1 - Float64(t_0 * -0.3333333333333333))
	t_3 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_4 = Float64(Float64(sin(x) * t_2) / cos(x))
	return Float64(Float64(Float64(eps * Float64(1.0 + t_3)) + Float64(Float64((eps ^ 2.0) * Float64(t_0 + t_1)) + Float64(Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(t_3 + t_4))) + Float64((eps ^ 4.0) * Float64(t_2 + Float64(Float64(Float64(sin(x) * Float64(t_4 - Float64(t_3 * -0.3333333333333333))) / cos(x)) + Float64(t_0 * 0.3333333333333333))))))) + fma(-1.0, tan(x), tan(x)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(t$95$0 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[x], $MachinePrecision] * t$95$2), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(eps * N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(t$95$2 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 - N[(t$95$3 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 62.1%

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

   1. tan-sum 62.3%

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

   2. div-inv 62.3%

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

   3. *-un-lft-identity 62.3%

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

   4. *-commutative 62.3%

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

   5. prod-diff 62.3%

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

   6. *-un-lft-identity 62.3%

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

   7. metadata-eval 62.3%

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

   8. *-un-lft-identity 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := 1 + t\_0\\ t_2 := {\tan x}^{2}\\ \mathsf{fma}\left(\varepsilon, t\_1, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{t\_1}}\right) + \left({\varepsilon}^{3} \cdot \left(\left(t\_0 \cdot t\_1 - \mathsf{fma}\left(-0.5, t\_1, t\_0 \cdot 0.16666666666666666\right)\right) - 0.16666666666666666\right) + {\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot \left(-1 - t\_0\right)\right) - \tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + t\_2, t\_2 \cdot 0.16666666666666666\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_1 (+ 1.0 t_0))
        (t_2 (pow (tan x) 2.0)))
   (+
    (fma eps t_1 (/ (pow eps 2.0) (/ (/ (cos x) (sin x)) t_1)))
    (+
     (*
      (pow eps 3.0)
      (-
       (- (* t_0 t_1) (fma -0.5 t_1 (* t_0 0.16666666666666666)))
       0.16666666666666666))
     (*
      (pow eps 4.0)
      (-
       (* -0.3333333333333333 (* (/ (sin x) (cos x)) (- -1.0 t_0)))
       (*
        (tan x)
        (+
         0.16666666666666666
         (-
          (fma -0.5 (+ 1.0 t_2) (* t_2 0.16666666666666666))
          (pow (* (tan x) (hypot 1.0 (tan x))) 2.0))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_1 = 1.0 + t_0;
	double t_2 = pow(tan(x), 2.0);
	return fma(eps, t_1, (pow(eps, 2.0) / ((cos(x) / sin(x)) / t_1))) + ((pow(eps, 3.0) * (((t_0 * t_1) - fma(-0.5, t_1, (t_0 * 0.16666666666666666))) - 0.16666666666666666)) + (pow(eps, 4.0) * ((-0.3333333333333333 * ((sin(x) / cos(x)) * (-1.0 - t_0))) - (tan(x) * (0.16666666666666666 + (fma(-0.5, (1.0 + t_2), (t_2 * 0.16666666666666666)) - pow((tan(x) * hypot(1.0, tan(x))), 2.0)))))));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_1 = Float64(1.0 + t_0)
	t_2 = tan(x) ^ 2.0
	return Float64(fma(eps, t_1, Float64((eps ^ 2.0) / Float64(Float64(cos(x) / sin(x)) / t_1))) + Float64(Float64((eps ^ 3.0) * Float64(Float64(Float64(t_0 * t_1) - fma(-0.5, t_1, Float64(t_0 * 0.16666666666666666))) - 0.16666666666666666)) + Float64((eps ^ 4.0) * Float64(Float64(-0.3333333333333333 * Float64(Float64(sin(x) / cos(x)) * Float64(-1.0 - t_0))) - Float64(tan(x) * Float64(0.16666666666666666 + Float64(fma(-0.5, Float64(1.0 + t_2), Float64(t_2 * 0.16666666666666666)) - (Float64(tan(x) * hypot(1.0, tan(x))) ^ 2.0))))))))
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]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(eps * t$95$1 + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] - N[(-0.5 * t$95$1 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Tan[x], $MachinePrecision] * N[(0.16666666666666666 + N[(N[(-0.5 * N[(1.0 + t$95$2), $MachinePrecision] + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)} \]
6. Step-by-step derivation

   1. tan-quot 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

   2. distribute-lft-in 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\left(\tan x \cdot 0.16666666666666666 + \tan x \cdot \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\left(\tan x \cdot 0.16666666666666666 + \tan x \cdot \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
8. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

10. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) + \left({\varepsilon}^{3} \cdot \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right)\right) - 0.16666666666666666\right) + {\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, {\tan x}^{2} \cdot 0.16666666666666666\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. tan-sum 62.3%

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

   2. div-inv 62.3%

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

   3. *-un-lft-identity 62.3%

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

   4. *-commutative 62.3%

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

   5. prod-diff 62.3%

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

   6. *-un-lft-identity 62.3%

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

   7. metadata-eval 62.3%

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

   8. *-un-lft-identity 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return fma(eps, (1.0 + (pow(sin(x), 2.0) * pow(cos(x), -2.0))), (pow(eps, 2.0) / (cos(x) / (sin(x) + (pow(sin(x), 3.0) / pow(cos(x), 2.0))))));
}

Julia

function code(x, eps)
	return fma(eps, Float64(1.0 + Float64((sin(x) ^ 2.0) * (cos(x) ^ -2.0))), Float64((eps ^ 2.0) / Float64(cos(x) / Float64(sin(x) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 2.0))))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. add-cube-cbrt 57.8%

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

   2. pow3 57.7%

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

4. Applied egg-rr 57.7%

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

   1. pow1/3 26.7%

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

6. Applied egg-rr 26.7%

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

   1. unpow1/3 57.7%

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

   2. rem-cube-cbrt 62.1%

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

   3. tan-quot 62.1%

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

   4. div-inv 62.1%

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

   5. *-commutative 62.1%

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

8. Applied egg-rr 62.1%

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

9. Taylor expanded in eps around 0 99.5%

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

    1. fma-def 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 6: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 7: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. cancel-sign-sub-inv 99.0%

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

   2. metadata-eval 99.0%

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

   3. *-lft-identity 99.0%

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

5. Simplified 99.0%

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

   1. distribute-rgt-in 99.0%

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

   2. *-un-lft-identity 99.0%

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

   3. add-sqr-sqrt 99.0%

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

   4. pow2 99.0%

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

   5. sqrt-div 99.0%

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

   6. unpow2 99.0%

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

   7. sqrt-prod 52.6%

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

   8. add-sqr-sqrt 99.0%

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

   9. unpow2 99.0%

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

   10. sqrt-prod 98.4%

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

   11. add-sqr-sqrt 99.0%

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

   12. tan-quot 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 8: 98.1% 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 62.1%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. cancel-sign-sub-inv 99.0%

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

   2. metadata-eval 99.0%

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

   3. *-lft-identity 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 98.0%

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

   1. *-commutative 98.0%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

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

Alternative 9: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in x around 0 97.5%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 97.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)} \]

   2. expm1-udef 7.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)} - 1} \]

   3. quot-tan 7.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\tan \varepsilon}\right)} - 1 \]

5. Applied egg-rr 7.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 97.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

   2. expm1-log1p 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 10: 97.7% 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 62.1%

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

3. Taylor expanded in x around 0 97.5%

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

4. Taylor expanded in eps around 0 97.5%

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

5. Final simplification 97.5%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(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)))

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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