2tan (problem 3.3.2)

Percentage Accurate: 41.7% → 99.5%

Time: 17.1s

Alternatives: 10

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: 41.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot t_2} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{\sin x}{\frac{\cos x}{t_2 \cdot -0.3333333333333333}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (/ (sin x) (cos x))))
   (if (<= eps -8.4e-5)
     (- (/ t_1 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 5.5e-5)
       (+
        (/ t_0 (- 1.0 (* t_0 t_2)))
        (+
         (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0)))
         (+
          (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
          (*
           (pow eps 3.0)
           (-
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
            (/ (sin x) (/ (cos x) (* t_2 -0.3333333333333333))))))))
       (-
        (* t_1 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double t_1 = tan(x) + tan(eps);
	double t_2 = sin(x) / cos(x);
	double tmp;
	if (eps <= -8.4e-5) {
		tmp = (t_1 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 5.5e-5) {
		tmp = (t_0 / (1.0 - (t_0 * t_2))) + ((eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0))) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) + (pow(eps, 3.0) * ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (sin(x) / (cos(x) / (t_2 * -0.3333333333333333)))))));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    t_1 = tan(x) + tan(eps)
    t_2 = sin(x) / cos(x)
    if (eps <= (-8.4d-5)) then
        tmp = (t_1 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 5.5d-5) then
        tmp = (t_0 / (1.0d0 - (t_0 * t_2))) + ((eps / ((cos(x) ** 2.0d0) / (sin(x) ** 2.0d0))) + (((eps * eps) / ((cos(x) ** 3.0d0) / (sin(x) ** 3.0d0))) + ((eps ** 3.0d0) * (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - (sin(x) / (cos(x) / (t_2 * (-0.3333333333333333d0))))))))
    else
        tmp = (t_1 * (1.0d0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) / Math.cos(eps);
	double t_1 = Math.tan(x) + Math.tan(eps);
	double t_2 = Math.sin(x) / Math.cos(x);
	double tmp;
	if (eps <= -8.4e-5) {
		tmp = (t_1 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 5.5e-5) {
		tmp = (t_0 / (1.0 - (t_0 * t_2))) + ((eps / (Math.pow(Math.cos(x), 2.0) / Math.pow(Math.sin(x), 2.0))) + (((eps * eps) / (Math.pow(Math.cos(x), 3.0) / Math.pow(Math.sin(x), 3.0))) + (Math.pow(eps, 3.0) * ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (Math.sin(x) / (Math.cos(x) / (t_2 * -0.3333333333333333)))))));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) / math.cos(eps)
	t_1 = math.tan(x) + math.tan(eps)
	t_2 = math.sin(x) / math.cos(x)
	tmp = 0
	if eps <= -8.4e-5:
		tmp = (t_1 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 5.5e-5:
		tmp = (t_0 / (1.0 - (t_0 * t_2))) + ((eps / (math.pow(math.cos(x), 2.0) / math.pow(math.sin(x), 2.0))) + (((eps * eps) / (math.pow(math.cos(x), 3.0) / math.pow(math.sin(x), 3.0))) + (math.pow(eps, 3.0) * ((math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (math.sin(x) / (math.cos(x) / (t_2 * -0.3333333333333333)))))))
	else:
		tmp = (t_1 * (1.0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	t_1 = Float64(tan(x) + tan(eps))
	t_2 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -8.4e-5)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 5.5e-5)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * t_2))) + Float64(Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) + Float64((eps ^ 3.0) * Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(sin(x) / Float64(cos(x) / Float64(t_2 * -0.3333333333333333))))))));
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) / cos(eps);
	t_1 = tan(x) + tan(eps);
	t_2 = sin(x) / cos(x);
	tmp = 0.0;
	if (eps <= -8.4e-5)
		tmp = (t_1 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 5.5e-5)
		tmp = (t_0 / (1.0 - (t_0 * t_2))) + ((eps / ((cos(x) ^ 2.0) / (sin(x) ^ 2.0))) + (((eps * eps) / ((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) + ((eps ^ 3.0) * (((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - (sin(x) / (cos(x) / (t_2 * -0.3333333333333333)))))));
	else
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -8.4e-5], N[(N[(t$95$1 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.5e-5], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(t$95$2 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -8.39999999999999954e-5

   1. Initial program 47.8%

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

      1. tan-sum 99.4%

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

      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. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -8.39999999999999954e-5 < eps < 5.5000000000000002e-5

   1. Initial program 27.2%

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

      1. tan-sum 28.2%

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

      2. div-inv 28.2%

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

   3. Applied egg-rr 28.2%

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

   4. Taylor expanded in x around inf 28.1%

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

      1. associate--l+ 59.0%

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

   6. Simplified 59.0%

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

   7. Taylor expanded in eps around 0 99.7%

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

      1. associate-/l* 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

   9. Simplified 99.7%

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

   if 5.5000000000000002e-5 < eps

   1. Initial program 55.6%

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

      1. tan-sum 99.4%

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

      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. Applied egg-rr 99.5%

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

      1. tan-quot 99.6%

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

      2. associate-*r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3e-7)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 2.4e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (-
        (* t_0 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3e-7) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 2.4e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(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 <= (-3d-7)) then
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 2.4d-7) then
        tmp = (eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = (t_0 * (1.0d0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(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 <= -3e-7) {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 2.4e-7) {
		tmp = (eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + ((eps * eps) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -3e-7:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 2.4e-7:
		tmp = (eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + ((eps * eps) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = (t_0 * (1.0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 2.4e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(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 <= -3e-7)
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 2.4e-7)
		tmp = (eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = (t_0 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(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, -3e-7], 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], If[LessEqual[eps, 2.4e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.9999999999999999e-7

   1. Initial program 48.6%

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

      1. tan-sum 99.4%

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

      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. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -2.9999999999999999e-7 < eps < 2.39999999999999979e-7

   1. Initial program 26.8%

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

      1. tan-sum 27.1%

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

      2. div-inv 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in eps around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

   6. Simplified 99.7%

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

   if 2.39999999999999979e-7 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. tan-quot 99.4%

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

      2. associate-*r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -1.9e-6)
     (- (/ t_1 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 8e-9)
       (+
        (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
        (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0))))
       (-
        (* t_1 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -1.9e-6) {
		tmp = (t_1 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 8e-9) {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + (eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(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) :: t_1
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    t_1 = tan(x) + tan(eps)
    if (eps <= (-1.9d-6)) then
        tmp = (t_1 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 8d-9) then
        tmp = (t_0 / (1.0d0 - (t_0 * (sin(x) / cos(x))))) + (eps / ((cos(x) ** 2.0d0) / (sin(x) ** 2.0d0)))
    else
        tmp = (t_1 * (1.0d0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) / Math.cos(eps);
	double t_1 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -1.9e-6) {
		tmp = (t_1 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 8e-9) {
		tmp = (t_0 / (1.0 - (t_0 * (Math.sin(x) / Math.cos(x))))) + (eps / (Math.pow(Math.cos(x), 2.0) / Math.pow(Math.sin(x), 2.0)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) / math.cos(eps)
	t_1 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -1.9e-6:
		tmp = (t_1 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 8e-9:
		tmp = (t_0 / (1.0 - (t_0 * (math.sin(x) / math.cos(x))))) + (eps / (math.pow(math.cos(x), 2.0) / math.pow(math.sin(x), 2.0)))
	else:
		tmp = (t_1 * (1.0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -1.9e-6)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 8e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) / cos(eps);
	t_1 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -1.9e-6)
		tmp = (t_1 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 8e-9)
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + (eps / ((cos(x) ^ 2.0) / (sin(x) ^ 2.0)));
	else
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.9e-6], N[(N[(t$95$1 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 8e-9], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.9e-6

   1. Initial program 47.8%

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

      1. tan-sum 99.4%

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

      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. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -1.9e-6 < eps < 8.0000000000000005e-9

   1. Initial program 27.3%

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

      1. tan-sum 27.7%

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

      2. div-inv 27.7%

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

   3. Applied egg-rr 27.7%

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

   4. Taylor expanded in x around inf 27.7%

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

      1. associate--l+ 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in eps around 0 99.6%

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

      1. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

   if 8.0000000000000005e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. tan-quot 99.4%

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

      2. associate-*r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.4e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 4e-9)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (-
        (* t_0 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.4e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 4e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(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 <= (-4.4d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 4d-9) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 * (1.0d0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(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 <= -4.4e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 4e-9) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -4.4e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 4e-9:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 * (1.0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.4e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 4e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(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 <= -4.4e-9)
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 4e-9)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 * (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(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, -4.4e-9], 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], If[LessEqual[eps, 4e-9], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.3999999999999997e-9

   1. Initial program 48.6%

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

      1. tan-sum 99.4%

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

      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. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -4.3999999999999997e-9 < eps < 4.00000000000000025e-9

   1. Initial program 26.8%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

   if 4.00000000000000025e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. tan-quot 99.4%

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

      2. associate-*r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -3.7e-9)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 1.7e-9)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* t_0 (/ 1.0 t_1)) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 1.7e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / 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) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    if (eps <= (-3.7d-9)) then
        tmp = (t_0 / t_1) - tan(x)
    else if (eps <= 1.7d-9) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 * (1.0d0 / t_1)) - 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 t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_0 / t_1) - Math.tan(x);
	} else if (eps <= 1.7e-9) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	tmp = 0
	if eps <= -3.7e-9:
		tmp = (t_0 / t_1) - math.tan(x)
	elif eps <= 1.7e-9:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 * (1.0 / t_1)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -3.7e-9)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 1.7e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = 1.0 - (tan(x) * tan(eps));
	tmp = 0.0;
	if (eps <= -3.7e-9)
		tmp = (t_0 / t_1) - tan(x);
	elseif (eps <= 1.7e-9)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 * (1.0 / t_1)) - 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]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.7e-9], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.7e-9], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.7e-9

   1. Initial program 48.6%

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

      1. tan-sum 99.4%

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

      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. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -3.7e-9 < eps < 1.6999999999999999e-9

   1. Initial program 26.8%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

   if 1.6999999999999999e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.4e-9) (not (<= eps 1.9e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.4e-9) || !(eps <= 1.9e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.4d-9)) .or. (.not. (eps <= 1.9d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.4e-9) || !(eps <= 1.9e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.4e-9) or not (eps <= 1.9e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.4e-9) || !(eps <= 1.9e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.4e-9) || ~((eps <= 1.9e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.4e-9], N[Not[LessEqual[eps, 1.9e-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[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.3999999999999997e-9 or 1.90000000000000006e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -4.3999999999999997e-9 < eps < 1.90000000000000006e-9

   1. Initial program 26.8%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.55e-6)
   (tan eps)
   (if (<= eps 9.2e-6)
     (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.55e-6) {
		tmp = tan(eps);
	} else if (eps <= 9.2e-6) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.55d-6)) then
        tmp = tan(eps)
    else if (eps <= 9.2d-6) then
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.55e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 9.2e-6) {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.55e-6:
		tmp = math.tan(eps)
	elif eps <= 9.2e-6:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.55e-6)
		tmp = tan(eps);
	elseif (eps <= 9.2e-6)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.55e-6)
		tmp = tan(eps);
	elseif (eps <= 9.2e-6)
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.55e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 9.2e-6], 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], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.55e-6 or 9.2e-6 < eps

   1. Initial program 52.1%

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

   2. Taylor expanded in x around 0 54.6%

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

      1. tan-quot 54.7%

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

      2. expm1-log1p-u 42.3%

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

      3. expm1-udef 41.6%

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

   4. Applied egg-rr 41.6%

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

      1. expm1-def 42.3%

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

      2. expm1-log1p 54.7%

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

   6. Simplified 54.7%

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

   if -1.55e-6 < eps < 9.2e-6

   1. Initial program 26.6%

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

      1. tan-sum 27.6%

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

      2. div-inv 27.6%

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

   3. Applied egg-rr 27.6%

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

      1. add-cube-cbrt 26.0%

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

      2. pow3 25.9%

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

   5. Applied egg-rr 25.9%

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

   6. Taylor expanded in eps around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. *-lft-identity 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.85e-6)
   (tan eps)
   (if (<= eps 4.7e-5)
     (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.85e-6) {
		tmp = tan(eps);
	} else if (eps <= 4.7e-5) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.85d-6)) then
        tmp = tan(eps)
    else if (eps <= 4.7d-5) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.85e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 4.7e-5) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.85e-6:
		tmp = math.tan(eps)
	elif eps <= 4.7e-5:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.85e-6)
		tmp = tan(eps);
	elseif (eps <= 4.7e-5)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.85e-6)
		tmp = tan(eps);
	elseif (eps <= 4.7e-5)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.85e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 4.7e-5], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.8500000000000001e-6 or 4.69999999999999972e-5 < eps

   1. Initial program 52.1%

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

   2. Taylor expanded in x around 0 54.6%

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

      1. tan-quot 54.7%

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

      2. expm1-log1p-u 42.3%

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

      3. expm1-udef 41.6%

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

   4. Applied egg-rr 41.6%

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

      1. expm1-def 42.3%

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

      2. expm1-log1p 54.7%

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

   6. Simplified 54.7%

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

   if -1.8500000000000001e-6 < eps < 4.69999999999999972e-5

   1. Initial program 26.6%

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

   2. Taylor expanded in eps around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. *-lft-identity 99.0%

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

      4. distribute-lft-in 99.1%

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

      5. *-rgt-identity 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 9: 57.8% 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 39.5%

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

2. Taylor expanded in x around 0 56.3%

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

   1. tan-quot 56.4%

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

   2. expm1-log1p-u 50.0%

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

   3. expm1-udef 24.1%

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

4. Applied egg-rr 24.1%

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

   1. expm1-def 50.0%

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

   2. expm1-log1p 56.4%

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

6. Simplified 56.4%

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

7. Final simplification 56.4%

   \[\leadsto \tan \varepsilon \]

Alternative 10: 31.5% 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 39.5%

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

2. Taylor expanded in x around 0 56.3%

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

3. Taylor expanded in eps around 0 30.8%

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

4. Final simplification 30.8%

   \[\leadsto \varepsilon \]

Developer target: 76.1% 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 2023196 
(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)))