2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.7%

Time: 28.5s

Alternatives: 7

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: 99.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)
    t_1 = sin(x) ** 2.0d0
    t_2 = t_1 / (cos(x) ** 2.0d0)
    t_3 = ((-0.3333333333333333d0) * t_2) - ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0))
    code = eps * (t_2 + ((eps * ((t_0 + (sin(x) / cos(x))) - (eps * ((t_3 - t_2) + ((eps * ((((-0.3333333333333333d0) * tan(x)) + (((-0.3333333333333333d0) * t_0) + ((sin(x) * t_3) / cos(x)))) - ((sin(x) * (0.3333333333333333d0 + (t_1 * (cos(x) ** (-2.0d0))))) / cos(x)))) - 0.3333333333333333d0))))) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. tan-sum 63.8%

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

   2. div-inv 63.8%

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

   3. fma-neg 63.8%

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

4. Applied egg-rr 63.8%

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

   1. fma-neg 63.8%

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

   2. associate-*r/ 63.8%

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

   3. *-rgt-identity 63.8%

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

6. Simplified 63.8%

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

7. Taylor expanded in eps around 0 100.0%

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

   1. tan-quot 100.0%

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

   2. pow1 100.0%

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

9. Applied egg-rr 100.0%

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

    1. unpow1 100.0%

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

11. Simplified 100.0%

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

    1. pow1 100.0%

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

    2. cancel-sign-sub-inv 100.0%

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

    3. metadata-eval 100.0%

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

    4. *-un-lft-identity 100.0%

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

    5. div-inv 100.0%

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

    6. pow-flip 100.0%

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

    7. metadata-eval 100.0%

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

13. Applied egg-rr 100.0%

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

    1. unpow1 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
   (+
    (*
     eps
     (+
      (* eps 0.3333333333333333)
      (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))))
    1.0))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + ((eps * ((eps * 0.3333333333333333) + ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + ((eps * ((eps * 0.3333333333333333d0) + (((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)) + (sin(x) / cos(x))))) + 1.0d0))
end function

Java

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

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + ((eps * ((eps * 0.3333333333333333) + ((math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)) + (math.sin(x) / math.cos(x))))) + 1.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(Float64(eps * Float64(Float64(eps * 0.3333333333333333) + Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x))))) + 1.0)))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. tan-sum 63.8%

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

   2. div-inv 63.8%

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

   3. fma-neg 63.8%

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

4. Applied egg-rr 63.8%

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

   1. fma-neg 63.8%

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

   2. associate-*r/ 63.8%

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

   3. *-rgt-identity 63.8%

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

6. Simplified 63.8%

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

7. Taylor expanded in eps around 0 100.0%

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

8. Taylor expanded in x around 0 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. sub-neg 99.0%

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

   2. mul-1-neg 99.0%

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

   3. remove-double-neg 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. Final simplification 99.0%

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

Alternative 6: 98.0% 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 63.7%

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

3. Taylor expanded in x around 0 98.0%

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

   1. tan-quot 98.0%

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

   2. *-un-lft-identity 98.0%

      \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]

5. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 98.0%

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

7. Simplified 98.0%

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

Alternative 7: 98.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in x around 0 98.0%

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

4. Taylor expanded in eps around 0 98.0%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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 2024087 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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