2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.7%

Time: 48.0s

Alternatives: 5

Speedup: 1.0×

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: 99.7% accurate, 0.0× speedup

Math

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

FPCore

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

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

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) / (cos(x) ** 2.0d0)
    t_1 = sin(x) / cos(x)
    t_2 = (sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)
    t_3 = ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - ((-0.3333333333333333d0) * t_0)
    code = ((eps ** 2.0d0) * (t_1 + t_2)) + (((eps ** 4.0d0) * (((sin(x) * (0.3333333333333333d0 + (tan(x) ** 2.0d0))) / cos(x)) + ((((sin(x) * t_3) / cos(x)) - (t_2 * (-0.3333333333333333d0))) - (t_1 * (-0.3333333333333333d0))))) + ((eps * (t_0 + 1.0d0)) + ((eps ** 3.0d0) * (0.3333333333333333d0 + (t_0 + t_3)))))
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);
	double t_1 = Math.sin(x) / Math.cos(x);
	double t_2 = Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0);
	double t_3 = (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (-0.3333333333333333 * t_0);
	return (Math.pow(eps, 2.0) * (t_1 + t_2)) + ((Math.pow(eps, 4.0) * (((Math.sin(x) * (0.3333333333333333 + Math.pow(Math.tan(x), 2.0))) / Math.cos(x)) + ((((Math.sin(x) * t_3) / Math.cos(x)) - (t_2 * -0.3333333333333333)) - (t_1 * -0.3333333333333333)))) + ((eps * (t_0 + 1.0)) + (Math.pow(eps, 3.0) * (0.3333333333333333 + (t_0 + t_3)))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	t_1 = math.sin(x) / math.cos(x)
	t_2 = math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)
	t_3 = (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (-0.3333333333333333 * t_0)
	return (math.pow(eps, 2.0) * (t_1 + t_2)) + ((math.pow(eps, 4.0) * (((math.sin(x) * (0.3333333333333333 + math.pow(math.tan(x), 2.0))) / math.cos(x)) + ((((math.sin(x) * t_3) / math.cos(x)) - (t_2 * -0.3333333333333333)) - (t_1 * -0.3333333333333333)))) + ((eps * (t_0 + 1.0)) + (math.pow(eps, 3.0) * (0.3333333333333333 + (t_0 + t_3)))))

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

MATLAB

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

Derivation

1. Initial program 63.8%

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

   1. tan-sum 64.0%

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

   2. clear-num 63.3%

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

4. Applied egg-rr 63.3%

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

5. Taylor expanded in eps around 0 99.9%

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

   1. expm1-log1p-u 99.9%

      \[\leadsto -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}}{\cos x} + \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right) \]

   2. expm1-udef 99.9%

      \[\leadsto -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-1 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} - 1}}{\cos x} + \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-1 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot \left(0.3333333333333333 + {\tan x}^{2}\right)\right)} - 1}}{\cos x} + \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-def 99.9%

      \[\leadsto -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(0.3333333333333333 + {\tan x}^{2}\right)\right)\right)}}{\cos x} + \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right) \]

   2. expm1-log1p 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. *-un-lft-identity 63.8%

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

   2. *-commutative 63.8%

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

   3. tan-quot 63.8%

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

   4. div-inv 63.7%

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

   5. prod-diff 63.7%

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

4. Applied egg-rr 63.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan \left(x + \varepsilon\right), 1, -\frac{1}{\cos x} \cdot \sin x\right) + \mathsf{fma}\left(-\frac{1}{\cos x}, \sin x, \frac{1}{\cos x} \cdot \sin x\right)} \]
5. Step-by-step derivation

   1. +-commutative 63.7%

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

   2. fma-udef 63.7%

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

   3. *-rgt-identity 63.7%

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

   4. associate-+r+ 63.7%

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

   5. unsub-neg 63.7%

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

6. Simplified 63.8%

   \[\leadsto \color{blue}{\left(0 + \tan \left(\varepsilon + x\right)\right) - \frac{\sin x}{\cos x}} \]
7. Step-by-step derivation

   1. +-lft-identity 63.8%

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

   2. +-commutative 63.8%

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

   3. tan-sum 63.9%

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

   4. tan-sum 63.8%

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

   5. tan-quot 63.8%

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

   6. frac-sub 63.8%

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

8. Applied egg-rr 63.8%

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

   1. *-commutative 63.8%

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

   2. fma-neg 63.8%

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

   3. distribute-rgt-neg-out 63.8%

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

   4. *-commutative 63.8%

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

   5. *-commutative 63.8%

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

10. Simplified 63.8%

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

11. Taylor expanded in x around 0 99.9%

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

12. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. *-un-lft-identity 63.8%

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

   2. *-commutative 63.8%

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

   3. tan-quot 63.8%

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

   4. div-inv 63.7%

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

   5. prod-diff 63.7%

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

4. Applied egg-rr 63.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan \left(x + \varepsilon\right), 1, -\frac{1}{\cos x} \cdot \sin x\right) + \mathsf{fma}\left(-\frac{1}{\cos x}, \sin x, \frac{1}{\cos x} \cdot \sin x\right)} \]
5. Step-by-step derivation

   1. +-commutative 63.7%

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

   2. fma-udef 63.7%

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

   3. *-rgt-identity 63.7%

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

   4. associate-+r+ 63.7%

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

   5. unsub-neg 63.7%

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

6. Simplified 63.8%

   \[\leadsto \color{blue}{\left(0 + \tan \left(\varepsilon + x\right)\right) - \frac{\sin x}{\cos x}} \]
7. Step-by-step derivation

   1. +-lft-identity 63.8%

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

   2. +-commutative 63.8%

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

   3. tan-sum 63.9%

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

   4. tan-sum 63.8%

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

   5. tan-quot 63.8%

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

   6. frac-sub 63.8%

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

8. Applied egg-rr 63.8%

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

   1. *-commutative 63.8%

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

   2. fma-neg 63.8%

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

   3. distribute-rgt-neg-out 63.8%

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

   4. *-commutative 63.8%

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

   5. *-commutative 63.8%

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

10. Simplified 63.8%

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

11. Taylor expanded in eps around 0 63.9%

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

    1. +-commutative 63.9%

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

    2. associate-+l+ 99.7%

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

    3. *-commutative 99.7%

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

    4. unpow2 99.7%

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

    5. unpow2 99.7%

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

    6. cos-sin-sum 99.7%

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

    7. *-lft-identity 99.7%

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

    8. distribute-rgt1-in 99.7%

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

    9. metadata-eval 99.7%

       \[\leadsto \frac{\varepsilon + \color{blue}{0} \cdot \left(\cos x \cdot \sin x\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

    10. mul0-lft 99.7%

        \[\leadsto \frac{\varepsilon + \color{blue}{0}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

13. Simplified 99.7%

    \[\leadsto \frac{\color{blue}{\varepsilon + 0}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

14. Final simplification 99.7%

    \[\leadsto \frac{\varepsilon}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
15. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon} \]
5. Add Preprocessing

Alternative 5: 5.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 63.8%

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

   1. add-cube-cbrt 59.4%

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

   2. pow3 59.3%

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

4. Applied egg-rr 59.3%

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

5. Taylor expanded in eps around 0 5.6%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \frac{\sin x}{\cos x} - \frac{\sin x}{\cos x}} \]
6. Step-by-step derivation

   1. pow-base-1 5.6%

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

   2. *-lft-identity 5.6%

      \[\leadsto \color{blue}{\frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x} \]

   3. +-inverses 5.6%

      \[\leadsto \color{blue}{0} \]

7. Simplified 5.6%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 5.6%

   \[\leadsto 0 \]
9. 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 2024031 
(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)))