2tan (problem 3.3.2)

Percentage Accurate: 41.9% → 99.5%

Time: 19.8s

Alternatives: 13

Speedup: 2.0×

Specification

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: 41.9% 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ t_2 := \frac{\tan \varepsilon}{t_1}\\ t_3 := {\cos x}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.033:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;t_2 + \left({\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - -0.3333333333333333 \cdot \frac{t_0}{t_3}\right) + \left(\frac{\varepsilon \cdot t_0}{t_3} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{\tan x}{t_1} - \tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- 1.0 (* (tan x) (tan eps))))
        (t_2 (/ (tan eps) t_1))
        (t_3 (pow (cos x) 2.0)))
   (if (<= eps -0.033)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
      (tan x))
     (if (<= eps 2.6e-9)
       (+
        t_2
        (+
         (*
          (pow eps 3.0)
          (-
           (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
           (* -0.3333333333333333 (/ t_0 t_3))))
         (+
          (/ (* eps t_0) t_3)
          (/ (* (pow eps 2.0) (pow (sin x) 3.0)) (pow (cos x) 3.0)))))
       (+ t_2 (- (/ (tan x) t_1) (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double t_2 = tan(eps) / t_1;
	double t_3 = pow(cos(x), 2.0);
	double tmp;
	if (eps <= -0.033) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = t_2 + ((pow(eps, 3.0) * ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (-0.3333333333333333 * (t_0 / t_3)))) + (((eps * t_0) / t_3) + ((pow(eps, 2.0) * pow(sin(x), 3.0)) / pow(cos(x), 3.0))));
	} else {
		tmp = t_2 + ((tan(x) / t_1) - tan(x));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    t_2 = tan(eps) / t_1
    t_3 = cos(x) ** 2.0d0
    if (eps <= (-0.033d0)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else if (eps <= 2.6d-9) then
        tmp = t_2 + (((eps ** 3.0d0) * (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - ((-0.3333333333333333d0) * (t_0 / t_3)))) + (((eps * t_0) / t_3) + (((eps ** 2.0d0) * (sin(x) ** 3.0d0)) / (cos(x) ** 3.0d0))))
    else
        tmp = t_2 + ((tan(x) / t_1) - tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double t_2 = Math.tan(eps) / t_1;
	double t_3 = Math.pow(Math.cos(x), 2.0);
	double tmp;
	if (eps <= -0.033) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = t_2 + ((Math.pow(eps, 3.0) * ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (-0.3333333333333333 * (t_0 / t_3)))) + (((eps * t_0) / t_3) + ((Math.pow(eps, 2.0) * Math.pow(Math.sin(x), 3.0)) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = t_2 + ((Math.tan(x) / t_1) - Math.tan(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	t_2 = math.tan(eps) / t_1
	t_3 = math.pow(math.cos(x), 2.0)
	tmp = 0
	if eps <= -0.033:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	elif eps <= 2.6e-9:
		tmp = t_2 + ((math.pow(eps, 3.0) * ((math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (-0.3333333333333333 * (t_0 / t_3)))) + (((eps * t_0) / t_3) + ((math.pow(eps, 2.0) * math.pow(math.sin(x), 3.0)) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = t_2 + ((math.tan(x) / t_1) - math.tan(x))
	return tmp

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_2 = Float64(tan(eps) / t_1)
	t_3 = cos(x) ^ 2.0
	tmp = 0.0
	if (eps <= -0.033)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	elseif (eps <= 2.6e-9)
		tmp = Float64(t_2 + Float64(Float64((eps ^ 3.0) * Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(-0.3333333333333333 * Float64(t_0 / t_3)))) + Float64(Float64(Float64(eps * t_0) / t_3) + Float64(Float64((eps ^ 2.0) * (sin(x) ^ 3.0)) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(t_2 + Float64(Float64(tan(x) / t_1) - tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = 1.0 - (tan(x) * tan(eps));
	t_2 = tan(eps) / t_1;
	t_3 = cos(x) ^ 2.0;
	tmp = 0.0;
	if (eps <= -0.033)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	elseif (eps <= 2.6e-9)
		tmp = t_2 + (((eps ^ 3.0) * (((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - (-0.3333333333333333 * (t_0 / t_3)))) + (((eps * t_0) / t_3) + (((eps ^ 2.0) * (sin(x) ^ 3.0)) / (cos(x) ^ 3.0))));
	else
		tmp = t_2 + ((tan(x) / t_1) - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[eps], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -0.033], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.6e-9], N[(t$95$2 + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * N[(t$95$0 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * t$95$0), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(N[Tan[x], $MachinePrecision] / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.033000000000000002

   1. Initial program 54.1%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. clear-num 99.6%

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

      4. tan-quot 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. clear-num 99.7%

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

      8. tan-quot 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-/l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.7%

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

   9. Simplified 99.7%

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

   if -0.033000000000000002 < eps < 2.6000000000000001e-9

   1. Initial program 29.9%

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

      1. tan-sum 30.5%

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

      2. div-inv 30.5%

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

      3. *-un-lft-identity 30.5%

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

      4. prod-diff 30.5%

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

      5. *-commutative 30.5%

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

      6. *-un-lft-identity 30.5%

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

      7. *-commutative 30.5%

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

      8. *-un-lft-identity 30.5%

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

   3. Applied egg-rr 30.5%

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

      1. +-commutative 30.5%

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

      2. fma-udef 30.5%

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

      3. associate-+r+ 30.5%

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

      4. unsub-neg 30.5%

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

   5. Simplified 30.5%

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

   6. Taylor expanded in x around inf 30.4%

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

      1. associate--l+ 63.2%

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

      2. associate-/r* 63.2%

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

      3. times-frac 63.2%

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

   8. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. tan-quot 63.2%

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

      3. tan-quot 63.2%

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

      4. tan-quot 63.2%

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

   10. Applied egg-rr 63.2%

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

   11. Taylor expanded in eps around 0 99.7%

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

   if 2.6000000000000001e-9 < eps

   1. Initial program 61.4%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

      2. associate-/r* 98.9%

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

      3. times-frac 98.9%

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

   8. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. tan-quot 99.5%

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

      3. tan-quot 99.5%

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

      4. tan-quot 99.5%

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

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.033:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left({\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \frac{{\sin x}^{2}}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.00032:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.0003:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + t_1\right) + {\varepsilon}^{3} \cdot \left(\cos x \cdot 0.3333333333333333 - -0.3333333333333333 \cdot t_1\right)}{\cos x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps))))
        (t_1 (/ (pow (sin x) 2.0) (cos x))))
   (if (<= eps -0.00032)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
      (tan x))
     (if (<= eps 0.0003)
       (/
        (+
         (* eps (+ (cos x) t_1))
         (*
          (pow eps 3.0)
          (- (* (cos x) 0.3333333333333333) (* -0.3333333333333333 t_1))))
        (* (cos x) t_0))
       (+ (/ (tan eps) t_0) (- (/ (tan x) t_0) (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = pow(sin(x), 2.0) / cos(x);
	double tmp;
	if (eps <= -0.00032) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else if (eps <= 0.0003) {
		tmp = ((eps * (cos(x) + t_1)) + (pow(eps, 3.0) * ((cos(x) * 0.3333333333333333) - (-0.3333333333333333 * t_1)))) / (cos(x) * t_0);
	} else {
		tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x));
	}
	return tmp;
}

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) :: tmp
    t_0 = 1.0d0 - (tan(x) * tan(eps))
    t_1 = (sin(x) ** 2.0d0) / cos(x)
    if (eps <= (-0.00032d0)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else if (eps <= 0.0003d0) then
        tmp = ((eps * (cos(x) + t_1)) + ((eps ** 3.0d0) * ((cos(x) * 0.3333333333333333d0) - ((-0.3333333333333333d0) * t_1)))) / (cos(x) * t_0)
    else
        tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double t_1 = Math.pow(Math.sin(x), 2.0) / Math.cos(x);
	double tmp;
	if (eps <= -0.00032) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else if (eps <= 0.0003) {
		tmp = ((eps * (Math.cos(x) + t_1)) + (Math.pow(eps, 3.0) * ((Math.cos(x) * 0.3333333333333333) - (-0.3333333333333333 * t_1)))) / (Math.cos(x) * t_0);
	} else {
		tmp = (Math.tan(eps) / t_0) + ((Math.tan(x) / t_0) - Math.tan(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(x) * math.tan(eps))
	t_1 = math.pow(math.sin(x), 2.0) / math.cos(x)
	tmp = 0
	if eps <= -0.00032:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	elif eps <= 0.0003:
		tmp = ((eps * (math.cos(x) + t_1)) + (math.pow(eps, 3.0) * ((math.cos(x) * 0.3333333333333333) - (-0.3333333333333333 * t_1)))) / (math.cos(x) * t_0)
	else:
		tmp = (math.tan(eps) / t_0) + ((math.tan(x) / t_0) - math.tan(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_1 = Float64((sin(x) ^ 2.0) / cos(x))
	tmp = 0.0
	if (eps <= -0.00032)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	elseif (eps <= 0.0003)
		tmp = Float64(Float64(Float64(eps * Float64(cos(x) + t_1)) + Float64((eps ^ 3.0) * Float64(Float64(cos(x) * 0.3333333333333333) - Float64(-0.3333333333333333 * t_1)))) / Float64(cos(x) * t_0));
	else
		tmp = Float64(Float64(tan(eps) / t_0) + Float64(Float64(tan(x) / t_0) - tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(x) * tan(eps));
	t_1 = (sin(x) ^ 2.0) / cos(x);
	tmp = 0.0;
	if (eps <= -0.00032)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	elseif (eps <= 0.0003)
		tmp = ((eps * (cos(x) + t_1)) + ((eps ^ 3.0) * ((cos(x) * 0.3333333333333333) - (-0.3333333333333333 * t_1)))) / (cos(x) * t_0);
	else
		tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00032], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0003], N[(N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(-0.3333333333333333 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.20000000000000026e-4

   1. Initial program 54.9%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

      8. *-un-lft-identity 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. clear-num 99.6%

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

      4. tan-quot 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. clear-num 99.7%

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

      8. tan-quot 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-/l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.7%

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

   9. Simplified 99.7%

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

   if -3.20000000000000026e-4 < eps < 2.99999999999999974e-4

   1. Initial program 30.8%

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

      1. tan-sum 31.9%

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

      2. tan-quot 31.6%

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

      3. frac-sub 31.6%

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

   3. Applied egg-rr 31.6%

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

   4. Taylor expanded in eps around 0 99.5%

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

   if 2.99999999999999974e-4 < eps

   1. Initial program 60.5%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.5%

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

      3. associate-+r+ 99.5%

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

      4. unsub-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf 99.1%

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

      1. associate--l+ 99.1%

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

      2. associate-/r* 99.1%

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

      3. times-frac 99.1%

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

   8. Simplified 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. tan-quot 99.7%

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

      3. tan-quot 99.7%

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

      4. tan-quot 99.7%

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

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00032:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.0003:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right) + {\varepsilon}^{3} \cdot \left(\cos x \cdot 0.3333333333333333 - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -0.033:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;t_1 + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps)))) (t_1 (/ (tan eps) t_0)))
   (if (<= eps -0.033)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
      (tan x))
     (if (<= eps 2.6e-9)
       (+
        t_1
        (+
         (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
         (/ (* (pow eps 2.0) (pow (sin x) 3.0)) (pow (cos x) 3.0))))
       (+ t_1 (- (/ (tan x) t_0) (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = tan(eps) / t_0;
	double tmp;
	if (eps <= -0.033) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = t_1 + (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + ((pow(eps, 2.0) * pow(sin(x), 3.0)) / pow(cos(x), 3.0)));
	} else {
		tmp = t_1 + ((tan(x) / t_0) - tan(x));
	}
	return tmp;
}

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) :: tmp
    t_0 = 1.0d0 - (tan(x) * tan(eps))
    t_1 = tan(eps) / t_0
    if (eps <= (-0.033d0)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else if (eps <= 2.6d-9) then
        tmp = t_1 + (((eps * (sin(x) ** 2.0d0)) / (cos(x) ** 2.0d0)) + (((eps ** 2.0d0) * (sin(x) ** 3.0d0)) / (cos(x) ** 3.0d0)))
    else
        tmp = t_1 + ((tan(x) / t_0) - tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double t_1 = Math.tan(eps) / t_0;
	double tmp;
	if (eps <= -0.033) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = t_1 + (((eps * Math.pow(Math.sin(x), 2.0)) / Math.pow(Math.cos(x), 2.0)) + ((Math.pow(eps, 2.0) * Math.pow(Math.sin(x), 3.0)) / Math.pow(Math.cos(x), 3.0)));
	} else {
		tmp = t_1 + ((Math.tan(x) / t_0) - Math.tan(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(x) * math.tan(eps))
	t_1 = math.tan(eps) / t_0
	tmp = 0
	if eps <= -0.033:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	elif eps <= 2.6e-9:
		tmp = t_1 + (((eps * math.pow(math.sin(x), 2.0)) / math.pow(math.cos(x), 2.0)) + ((math.pow(eps, 2.0) * math.pow(math.sin(x), 3.0)) / math.pow(math.cos(x), 3.0)))
	else:
		tmp = t_1 + ((math.tan(x) / t_0) - math.tan(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_1 = Float64(tan(eps) / t_0)
	tmp = 0.0
	if (eps <= -0.033)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	elseif (eps <= 2.6e-9)
		tmp = Float64(t_1 + Float64(Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)) + Float64(Float64((eps ^ 2.0) * (sin(x) ^ 3.0)) / (cos(x) ^ 3.0))));
	else
		tmp = Float64(t_1 + Float64(Float64(tan(x) / t_0) - tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(x) * tan(eps));
	t_1 = tan(eps) / t_0;
	tmp = 0.0;
	if (eps <= -0.033)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	elseif (eps <= 2.6e-9)
		tmp = t_1 + (((eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)) + (((eps ^ 2.0) * (sin(x) ^ 3.0)) / (cos(x) ^ 3.0)));
	else
		tmp = t_1 + ((tan(x) / t_0) - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[eps, -0.033], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.6e-9], N[(t$95$1 + N[(N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.033000000000000002

   1. Initial program 54.1%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. clear-num 99.6%

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

      4. tan-quot 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. clear-num 99.7%

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

      8. tan-quot 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-/l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.7%

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

   9. Simplified 99.7%

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

   if -0.033000000000000002 < eps < 2.6000000000000001e-9

   1. Initial program 29.9%

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

      1. tan-sum 30.5%

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

      2. div-inv 30.5%

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

      3. *-un-lft-identity 30.5%

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

      4. prod-diff 30.5%

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

      5. *-commutative 30.5%

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

      6. *-un-lft-identity 30.5%

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

      7. *-commutative 30.5%

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

      8. *-un-lft-identity 30.5%

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

   3. Applied egg-rr 30.5%

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

      1. +-commutative 30.5%

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

      2. fma-udef 30.5%

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

      3. associate-+r+ 30.5%

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

      4. unsub-neg 30.5%

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

   5. Simplified 30.5%

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

   6. Taylor expanded in x around inf 30.4%

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

      1. associate--l+ 63.2%

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

      2. associate-/r* 63.2%

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

      3. times-frac 63.2%

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

   8. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. tan-quot 63.2%

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

      3. tan-quot 63.2%

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

      4. tan-quot 63.2%

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

   10. Applied egg-rr 63.2%

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

   11. Taylor expanded in eps around 0 99.5%

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

   if 2.6000000000000001e-9 < eps

   1. Initial program 61.4%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

      2. associate-/r* 98.9%

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

      3. times-frac 98.9%

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

   8. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. tan-quot 99.5%

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

      3. tan-quot 99.5%

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

      4. tan-quot 99.5%

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

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.033:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot t_1 + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot t_1\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps))))
        (t_1 (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0)))
   (if (<= eps -2.1e-6)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
      (tan x))
     (if (<= eps 2.6e-9)
       (+ (* eps t_1) (/ (* (pow eps 2.0) (* (sin x) t_1)) (cos x)))
       (+ (/ (tan eps) t_0) (- (/ (tan x) t_0) (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = (pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0;
	double tmp;
	if (eps <= -2.1e-6) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = (eps * t_1) + ((pow(eps, 2.0) * (sin(x) * t_1)) / cos(x));
	} else {
		tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x));
	}
	return tmp;
}

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) :: tmp
    t_0 = 1.0d0 - (tan(x) * tan(eps))
    t_1 = ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0
    if (eps <= (-2.1d-6)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else if (eps <= 2.6d-9) then
        tmp = (eps * t_1) + (((eps ** 2.0d0) * (sin(x) * t_1)) / cos(x))
    else
        tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double t_1 = (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0;
	double tmp;
	if (eps <= -2.1e-6) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = (eps * t_1) + ((Math.pow(eps, 2.0) * (Math.sin(x) * t_1)) / Math.cos(x));
	} else {
		tmp = (Math.tan(eps) / t_0) + ((Math.tan(x) / t_0) - Math.tan(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(x) * math.tan(eps))
	t_1 = (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0
	tmp = 0
	if eps <= -2.1e-6:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	elif eps <= 2.6e-9:
		tmp = (eps * t_1) + ((math.pow(eps, 2.0) * (math.sin(x) * t_1)) / math.cos(x))
	else:
		tmp = (math.tan(eps) / t_0) + ((math.tan(x) / t_0) - math.tan(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_1 = Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (eps <= -2.1e-6)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	elseif (eps <= 2.6e-9)
		tmp = Float64(Float64(eps * t_1) + Float64(Float64((eps ^ 2.0) * Float64(sin(x) * t_1)) / cos(x)));
	else
		tmp = Float64(Float64(tan(eps) / t_0) + Float64(Float64(tan(x) / t_0) - tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(x) * tan(eps));
	t_1 = ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0;
	tmp = 0.0;
	if (eps <= -2.1e-6)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	elseif (eps <= 2.6e-9)
		tmp = (eps * t_1) + (((eps ^ 2.0) * (sin(x) * t_1)) / cos(x));
	else
		tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[eps, -2.1e-6], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.6e-9], N[(N[(eps * t$95$1), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.0999999999999998e-6

   1. Initial program 54.9%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

      8. *-un-lft-identity 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. clear-num 99.6%

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

      4. tan-quot 99.7%

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

      5. frac-times 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. clear-num 99.7%

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

      8. tan-quot 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-/l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.7%

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

   9. Simplified 99.7%

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

   if -2.0999999999999998e-6 < eps < 2.6000000000000001e-9

   1. Initial program 29.4%

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

   2. Taylor expanded in eps around 0 99.4%

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

   if 2.6000000000000001e-9 < eps

   1. Initial program 61.4%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

      2. associate-/r* 98.9%

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

      3. times-frac 98.9%

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

   8. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. tan-quot 99.5%

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

      3. tan-quot 99.5%

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

      4. tan-quot 99.5%

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

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -4.1e-9)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
      (tan x))
     (if (<= eps 2.4e-9)
       (/
        eps
        (/ (cos x) (+ (cos x) (/ (- 0.5 (* 0.5 (cos (* x 2.0)))) (cos x)))))
       (+ (/ (tan eps) t_0) (- (/ (tan x) t_0) (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -4.1e-9) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else if (eps <= 2.4e-9) {
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	} else {
		tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(x) * tan(eps))
    if (eps <= (-4.1d-9)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else if (eps <= 2.4d-9) then
        tmp = eps / (cos(x) / (cos(x) + ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) / cos(x))))
    else
        tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double tmp;
	if (eps <= -4.1e-9) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else if (eps <= 2.4e-9) {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + ((0.5 - (0.5 * Math.cos((x * 2.0)))) / Math.cos(x))));
	} else {
		tmp = (Math.tan(eps) / t_0) + ((Math.tan(x) / t_0) - Math.tan(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(x) * math.tan(eps))
	tmp = 0
	if eps <= -4.1e-9:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	elif eps <= 2.4e-9:
		tmp = eps / (math.cos(x) / (math.cos(x) + ((0.5 - (0.5 * math.cos((x * 2.0)))) / math.cos(x))))
	else:
		tmp = (math.tan(eps) / t_0) + ((math.tan(x) / t_0) - math.tan(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -4.1e-9)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	elseif (eps <= 2.4e-9)
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))) / cos(x)))));
	else
		tmp = Float64(Float64(tan(eps) / t_0) + Float64(Float64(tan(x) / t_0) - tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(x) * tan(eps));
	tmp = 0.0;
	if (eps <= -4.1e-9)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	elseif (eps <= 2.4e-9)
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	else
		tmp = (tan(eps) / t_0) + ((tan(x) / t_0) - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.1e-9], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.4e-9], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.1000000000000003e-9

   1. Initial program 54.0%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-udef 99.1%

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

      3. associate-+r+ 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. tan-quot 99.2%

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

      3. clear-num 99.1%

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

      4. tan-quot 99.3%

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

      5. frac-times 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. clear-num 99.3%

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

      8. tan-quot 99.2%

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

   7. Applied egg-rr 99.2%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. associate-/l* 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/l* 99.2%

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

   9. Simplified 99.2%

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

   if -4.1000000000000003e-9 < eps < 2.4e-9

   1. Initial program 29.6%

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

      1. tan-sum 29.6%

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

      2. tan-quot 29.3%

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

      3. frac-sub 29.3%

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

   3. Applied egg-rr 29.3%

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

   4. Taylor expanded in eps around 0 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-sub 99.6%

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

      2. +-inverses 99.6%

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

      3. cos-0 99.6%

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

      4. metadata-eval 99.6%

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

      5. count-2 99.6%

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

      6. *-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/l* 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 2.4e-9 < eps

   1. Initial program 61.4%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

      2. associate-/r* 98.9%

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

      3. times-frac 98.9%

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

   8. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. tan-quot 99.5%

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

      3. tan-quot 99.5%

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

      4. tan-quot 99.5%

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

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.9e-9) (not (<= eps 2.6e-9)))
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
    (tan x))
   (/
    eps
    (/ (cos x) (+ (cos x) (/ (- 0.5 (* 0.5 (cos (* x 2.0)))) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.9e-9) || !(eps <= 2.6e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else {
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.9d-9)) .or. (.not. (eps <= 2.6d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else
        tmp = eps / (cos(x) / (cos(x) + ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) / cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.9e-9) || !(eps <= 2.6e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + ((0.5 - (0.5 * Math.cos((x * 2.0)))) / Math.cos(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.9e-9) or not (eps <= 2.6e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	else:
		tmp = eps / (math.cos(x) / (math.cos(x) + ((0.5 - (0.5 * math.cos((x * 2.0)))) / math.cos(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.9e-9) || !(eps <= 2.6e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	else
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))) / cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.9e-9) || ~((eps <= 2.6e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	else
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.9e-9], N[Not[LessEqual[eps, 2.6e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.89999999999999991e-9 or 2.6000000000000001e-9 < eps

   1. Initial program 57.9%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. tan-quot 99.2%

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

      3. clear-num 99.2%

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

      4. tan-quot 99.3%

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

      5. frac-times 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. clear-num 99.3%

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

      8. tan-quot 99.3%

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

   7. Applied egg-rr 99.3%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. associate-/l* 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/l* 99.3%

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

   9. Simplified 99.3%

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

   if -2.89999999999999991e-9 < eps < 2.6000000000000001e-9

   1. Initial program 29.6%

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

      1. tan-sum 29.6%

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

      2. tan-quot 29.3%

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

      3. frac-sub 29.3%

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

   3. Applied egg-rr 29.3%

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

   4. Taylor expanded in eps around 0 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-sub 99.6%

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

      2. +-inverses 99.6%

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

      3. cos-0 99.6%

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

      4. metadata-eval 99.6%

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

      5. count-2 99.6%

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

      6. *-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/l* 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{1 \cdot \frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - \cos \left(x \cdot 2\right) \cdot 0.5}{\cos x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.3e-9) (not (<= eps 2.6e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (/
    eps
    (/ (cos x) (+ (cos x) (/ (- 0.5 (* 0.5 (cos (* x 2.0)))) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-9) || !(eps <= 2.6e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.3d-9)) .or. (.not. (eps <= 2.6d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps / (cos(x) / (cos(x) + ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) / cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-9) || !(eps <= 2.6e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + ((0.5 - (0.5 * Math.cos((x * 2.0)))) / Math.cos(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.3e-9) or not (eps <= 2.6e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps / (math.cos(x) / (math.cos(x) + ((0.5 - (0.5 * math.cos((x * 2.0)))) / math.cos(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.3e-9) || !(eps <= 2.6e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))) / cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.3e-9) || ~((eps <= 2.6e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.3e-9], N[Not[LessEqual[eps, 2.6e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.30000000000000018e-9 or 2.6000000000000001e-9 < eps

   1. Initial program 57.9%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.2%

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

   if -3.30000000000000018e-9 < eps < 2.6000000000000001e-9

   1. Initial program 29.6%

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

      1. tan-sum 29.6%

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

      2. tan-quot 29.3%

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

      3. frac-sub 29.3%

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

   3. Applied egg-rr 29.3%

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

   4. Taylor expanded in eps around 0 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-sub 99.6%

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

      2. +-inverses 99.6%

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

      3. cos-0 99.6%

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

      4. metadata-eval 99.6%

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

      5. count-2 99.6%

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

      6. *-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/l* 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{1 \cdot \frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - \cos \left(x \cdot 2\right) \cdot 0.5}{\cos x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \end{array} \]

Alternative 8: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2.95e-9)
     (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 2e-9)
       (/
        eps
        (/ (cos x) (+ (cos x) (/ (- 0.5 (* 0.5 (cos (* x 2.0)))) (cos x)))))
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.95e-9) {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 2e-9) {
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-2.95d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    else if (eps <= 2d-9) then
        tmp = eps / (cos(x) / (cos(x) + ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) / cos(x))))
    else
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -2.95e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	} else if (eps <= 2e-9) {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + ((0.5 - (0.5 * Math.cos((x * 2.0)))) / Math.cos(x))));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -2.95e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	elif eps <= 2e-9:
		tmp = eps / (math.cos(x) / (math.cos(x) + ((0.5 - (0.5 * math.cos((x * 2.0)))) / math.cos(x))))
	else:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.95e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 2e-9)
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))) / cos(x)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -2.95e-9)
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	elseif (eps <= 2e-9)
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	else
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.95e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2e-9], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.9499999999999999e-9

   1. Initial program 54.0%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-udef 99.1%

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

      3. associate-+r+ 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. tan-quot 99.2%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.2%

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

      4. clear-num 99.2%

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

      5. tan-quot 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -2.9499999999999999e-9 < eps < 2.00000000000000012e-9

   1. Initial program 29.6%

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

      1. tan-sum 29.6%

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

      2. tan-quot 29.3%

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

      3. frac-sub 29.3%

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

   3. Applied egg-rr 29.3%

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

   4. Taylor expanded in eps around 0 99.5%

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

      1. unpow2 99.5%

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

      2. sin-mult 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. div-sub 99.6%

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

      2. +-inverses 99.6%

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

      3. cos-0 99.6%

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

      4. metadata-eval 99.6%

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

      5. count-2 99.6%

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

      6. *-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/l* 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 2.00000000000000012e-9 < eps

   1. Initial program 61.4%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 9: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{\cos x + \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{\cos x}}{\cos x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -6.4e-6) (not (<= eps 0.000145)))
   (tan eps)
   (*
    eps
    (/ (+ (cos x) (/ (+ 0.5 (* (cos (* x 2.0)) -0.5)) (cos x))) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.4e-6) || !(eps <= 0.000145)) {
		tmp = tan(eps);
	} else {
		tmp = eps * ((cos(x) + ((0.5 + (cos((x * 2.0)) * -0.5)) / cos(x))) / cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-6.4d-6)) .or. (.not. (eps <= 0.000145d0))) then
        tmp = tan(eps)
    else
        tmp = eps * ((cos(x) + ((0.5d0 + (cos((x * 2.0d0)) * (-0.5d0))) / cos(x))) / cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.4e-6) || !(eps <= 0.000145)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * ((Math.cos(x) + ((0.5 + (Math.cos((x * 2.0)) * -0.5)) / Math.cos(x))) / Math.cos(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -6.4e-6) or not (eps <= 0.000145):
		tmp = math.tan(eps)
	else:
		tmp = eps * ((math.cos(x) + ((0.5 + (math.cos((x * 2.0)) * -0.5)) / math.cos(x))) / math.cos(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -6.4e-6) || !(eps <= 0.000145))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(Float64(cos(x) + Float64(Float64(0.5 + Float64(cos(Float64(x * 2.0)) * -0.5)) / cos(x))) / cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -6.4e-6) || ~((eps <= 0.000145)))
		tmp = tan(eps);
	else
		tmp = eps * ((cos(x) + ((0.5 + (cos((x * 2.0)) * -0.5)) / cos(x))) / cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -6.4e-6], N[Not[LessEqual[eps, 0.000145]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 + N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.3999999999999997e-6 or 1.45e-4 < eps

   1. Initial program 57.8%

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

   2. Taylor expanded in x around 0 60.8%

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

      1. tan-quot 61.1%

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

      2. expm1-log1p-u 44.7%

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

      3. expm1-udef 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. expm1-def 44.7%

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

      2. expm1-log1p 61.1%

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

   6. Simplified 61.1%

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

   if -6.3999999999999997e-6 < eps < 1.45e-4

   1. Initial program 30.8%

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

      1. tan-sum 31.9%

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

      2. tan-quot 31.6%

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

      3. frac-sub 31.6%

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

   3. Applied egg-rr 31.6%

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

   4. Taylor expanded in eps around 0 98.7%

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

      1. unpow2 98.7%

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

      2. sin-mult 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. div-sub 98.7%

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

      2. +-inverses 98.7%

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

      3. cos-0 98.7%

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

      4. metadata-eval 98.7%

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

      5. count-2 98.7%

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

      6. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in eps around 0 98.7%

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

       1. sub-neg 98.7%

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

       2. associate-*r/ 98.7%

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

       3. *-commutative 98.7%

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

       4. *-commutative 98.7%

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

       5. associate-*r/ 98.7%

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

       6. neg-mul-1 98.7%

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

       7. remove-double-neg 98.7%

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

       8. associate-*r/ 98.7%

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

   11. Simplified 98.7%

       \[\leadsto \color{blue}{\varepsilon \cdot \frac{\cos x + \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{\cos x}}{\cos x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{\cos x + \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{\cos x}}{\cos x}\\ \end{array} \]

Alternative 10: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.3e-5) (not (<= eps 0.000145)))
   (tan eps)
   (/
    eps
    (/ (cos x) (+ (cos x) (/ (- 0.5 (* 0.5 (cos (* x 2.0)))) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.3e-5) || !(eps <= 0.000145)) {
		tmp = tan(eps);
	} else {
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.3d-5)) .or. (.not. (eps <= 0.000145d0))) then
        tmp = tan(eps)
    else
        tmp = eps / (cos(x) / (cos(x) + ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) / cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.3e-5) || !(eps <= 0.000145)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + ((0.5 - (0.5 * Math.cos((x * 2.0)))) / Math.cos(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.3e-5) or not (eps <= 0.000145):
		tmp = math.tan(eps)
	else:
		tmp = eps / (math.cos(x) / (math.cos(x) + ((0.5 - (0.5 * math.cos((x * 2.0)))) / math.cos(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.3e-5) || !(eps <= 0.000145))
		tmp = tan(eps);
	else
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))) / cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.3e-5) || ~((eps <= 0.000145)))
		tmp = tan(eps);
	else
		tmp = eps / (cos(x) / (cos(x) + ((0.5 - (0.5 * cos((x * 2.0)))) / cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.3e-5], N[Not[LessEqual[eps, 0.000145]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.29999999999999992e-5 or 1.45e-4 < eps

   1. Initial program 57.8%

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

   2. Taylor expanded in x around 0 60.8%

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

      1. tan-quot 61.1%

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

      2. expm1-log1p-u 44.7%

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

      3. expm1-udef 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. expm1-def 44.7%

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

      2. expm1-log1p 61.1%

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

   6. Simplified 61.1%

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

   if -1.29999999999999992e-5 < eps < 1.45e-4

   1. Initial program 30.8%

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

      1. tan-sum 31.9%

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

      2. tan-quot 31.6%

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

      3. frac-sub 31.6%

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

   3. Applied egg-rr 31.6%

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

   4. Taylor expanded in eps around 0 98.7%

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

      1. unpow2 98.7%

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

      2. sin-mult 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. div-sub 98.7%

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

      2. +-inverses 98.7%

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

      3. cos-0 98.7%

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

      4. metadata-eval 98.7%

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

      5. count-2 98.7%

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

      6. *-commutative 98.7%

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

   8. Simplified 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. associate-/l* 98.7%

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

      3. cancel-sign-sub-inv 98.7%

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

      4. metadata-eval 98.7%

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

      5. *-un-lft-identity 98.7%

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

      6. div-inv 98.7%

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

      7. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - \cos \left(x \cdot 2\right) \cdot 0.5}{\cos x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{\cos x}}}\\ \end{array} \]

Alternative 11: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.3e-6) (not (<= eps 0.000145)))
   (tan eps)
   (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-6) || !(eps <= 0.000145)) {
		tmp = tan(eps);
	} else {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.3d-6)) .or. (.not. (eps <= 0.000145d0))) then
        tmp = tan(eps)
    else
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-6) || !(eps <= 0.000145)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.3e-6) or not (eps <= 0.000145):
		tmp = math.tan(eps)
	else:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.3e-6) || !(eps <= 0.000145))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.3e-6) || ~((eps <= 0.000145)))
		tmp = tan(eps);
	else
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.3e-6], N[Not[LessEqual[eps, 0.000145]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], 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. Split input into 2 regimes

2. if eps < -3.30000000000000017e-6 or 1.45e-4 < eps

   1. Initial program 57.8%

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

   2. Taylor expanded in x around 0 60.8%

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

      1. tan-quot 61.1%

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

      2. expm1-log1p-u 44.7%

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

      3. expm1-udef 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. expm1-def 44.7%

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

      2. expm1-log1p 61.1%

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

   6. Simplified 61.1%

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

   if -3.30000000000000017e-6 < eps < 1.45e-4

   1. Initial program 30.8%

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

   2. Taylor expanded in eps around 0 98.7%

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

      1. cancel-sign-sub-inv 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

   4. Simplified 98.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \]

Alternative 12: 57.3% 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 43.8%

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

2. Taylor expanded in x around 0 61.9%

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

   1. tan-quot 62.1%

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

   2. expm1-log1p-u 54.2%

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

   3. expm1-udef 24.9%

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

4. Applied egg-rr 24.9%

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

   1. expm1-def 54.2%

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

   2. expm1-log1p 62.1%

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

6. Simplified 62.1%

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

7. Final simplification 62.1%

   \[\leadsto \tan \varepsilon \]

Alternative 13: 31.1% 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 43.8%

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

2. Taylor expanded in x around 0 61.9%

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

3. Taylor expanded in eps around 0 34.4%

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

4. Final simplification 34.4%

   \[\leadsto \varepsilon \]

Developer target: 75.8% 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 2023321 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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