2tan (problem 3.3.2)

Percentage Accurate: 42.6% → 99.6%

Time: 20.3s

Alternatives: 13

Speedup: 1.9×

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: 42.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_3 := \frac{\sin x}{\cos x}\\ t_4 := t_3 \cdot -0.3333333333333333\\ t_5 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ \mathbf{if}\;\varepsilon \leq -0.00037 \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{1 - t_2 \cdot t_3} + \tan x \cdot \left(t_0 \cdot \frac{\varepsilon \cdot \varepsilon}{t_1} + \left(\left(\frac{\varepsilon \cdot \sin x}{\cos x} - {\varepsilon}^{3} \cdot \left(t_4 - t_5\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot \frac{t_5 - t_4}{\cos x} - -0.3333333333333333 \cdot \frac{t_0}{t_1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ (sin eps) (cos eps)))
        (t_3 (/ (sin x) (cos x)))
        (t_4 (* t_3 -0.3333333333333333))
        (t_5 (/ (pow (sin x) 3.0) (pow (cos x) 3.0))))
   (if (or (<= eps -0.00037) (not (<= eps 8.2e-9)))
     (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (+
      (/ t_2 (- 1.0 (* t_2 t_3)))
      (*
       (tan x)
       (+
        (* t_0 (/ (* eps eps) t_1))
        (+
         (- (/ (* eps (sin x)) (cos x)) (* (pow eps 3.0) (- t_4 t_5)))
         (*
          (pow eps 4.0)
          (-
           (* (sin x) (/ (- t_5 t_4) (cos x)))
           (* -0.3333333333333333 (/ t_0 t_1)))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = sin(eps) / cos(eps);
	double t_3 = sin(x) / cos(x);
	double t_4 = t_3 * -0.3333333333333333;
	double t_5 = pow(sin(x), 3.0) / pow(cos(x), 3.0);
	double tmp;
	if ((eps <= -0.00037) || !(eps <= 8.2e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = (t_2 / (1.0 - (t_2 * t_3))) + (tan(x) * ((t_0 * ((eps * eps) / t_1)) + ((((eps * sin(x)) / cos(x)) - (pow(eps, 3.0) * (t_4 - t_5))) + (pow(eps, 4.0) * ((sin(x) * ((t_5 - t_4) / cos(x))) - (-0.3333333333333333 * (t_0 / t_1)))))));
	}
	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) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = sin(eps) / cos(eps)
    t_3 = sin(x) / cos(x)
    t_4 = t_3 * (-0.3333333333333333d0)
    t_5 = (sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)
    if ((eps <= (-0.00037d0)) .or. (.not. (eps <= 8.2d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = (t_2 / (1.0d0 - (t_2 * t_3))) + (tan(x) * ((t_0 * ((eps * eps) / t_1)) + ((((eps * sin(x)) / cos(x)) - ((eps ** 3.0d0) * (t_4 - t_5))) + ((eps ** 4.0d0) * ((sin(x) * ((t_5 - t_4) / cos(x))) - ((-0.3333333333333333d0) * (t_0 / t_1)))))))
    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 = Math.pow(Math.cos(x), 2.0);
	double t_2 = Math.sin(eps) / Math.cos(eps);
	double t_3 = Math.sin(x) / Math.cos(x);
	double t_4 = t_3 * -0.3333333333333333;
	double t_5 = Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0);
	double tmp;
	if ((eps <= -0.00037) || !(eps <= 8.2e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = (t_2 / (1.0 - (t_2 * t_3))) + (Math.tan(x) * ((t_0 * ((eps * eps) / t_1)) + ((((eps * Math.sin(x)) / Math.cos(x)) - (Math.pow(eps, 3.0) * (t_4 - t_5))) + (Math.pow(eps, 4.0) * ((Math.sin(x) * ((t_5 - t_4) / Math.cos(x))) - (-0.3333333333333333 * (t_0 / t_1)))))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = math.sin(eps) / math.cos(eps)
	t_3 = math.sin(x) / math.cos(x)
	t_4 = t_3 * -0.3333333333333333
	t_5 = math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)
	tmp = 0
	if (eps <= -0.00037) or not (eps <= 8.2e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = (t_2 / (1.0 - (t_2 * t_3))) + (math.tan(x) * ((t_0 * ((eps * eps) / t_1)) + ((((eps * math.sin(x)) / math.cos(x)) - (math.pow(eps, 3.0) * (t_4 - t_5))) + (math.pow(eps, 4.0) * ((math.sin(x) * ((t_5 - t_4) / math.cos(x))) - (-0.3333333333333333 * (t_0 / t_1)))))))
	return tmp

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(sin(eps) / cos(eps))
	t_3 = Float64(sin(x) / cos(x))
	t_4 = Float64(t_3 * -0.3333333333333333)
	t_5 = Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))
	tmp = 0.0
	if ((eps <= -0.00037) || !(eps <= 8.2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(Float64(t_2 / Float64(1.0 - Float64(t_2 * t_3))) + Float64(tan(x) * Float64(Float64(t_0 * Float64(Float64(eps * eps) / t_1)) + Float64(Float64(Float64(Float64(eps * sin(x)) / cos(x)) - Float64((eps ^ 3.0) * Float64(t_4 - t_5))) + Float64((eps ^ 4.0) * Float64(Float64(sin(x) * Float64(Float64(t_5 - t_4) / cos(x))) - Float64(-0.3333333333333333 * Float64(t_0 / t_1))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = sin(eps) / cos(eps);
	t_3 = sin(x) / cos(x);
	t_4 = t_3 * -0.3333333333333333;
	t_5 = (sin(x) ^ 3.0) / (cos(x) ^ 3.0);
	tmp = 0.0;
	if ((eps <= -0.00037) || ~((eps <= 8.2e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = (t_2 / (1.0 - (t_2 * t_3))) + (tan(x) * ((t_0 * ((eps * eps) / t_1)) + ((((eps * sin(x)) / cos(x)) - ((eps ^ 3.0) * (t_4 - t_5))) + ((eps ^ 4.0) * ((sin(x) * ((t_5 - t_4) / cos(x))) - (-0.3333333333333333 * (t_0 / t_1)))))));
	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[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * -0.3333333333333333), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.00037], N[Not[LessEqual[eps, 8.2e-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[(N[(t$95$2 / N[(1.0 - N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(t$95$0 * N[(N[(eps * eps), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, 3.0], $MachinePrecision] * N[(t$95$4 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$5 - t$95$4), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6999999999999999e-4 or 8.2000000000000006e-9 < eps

   1. Initial program 49.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

   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 99.3%

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

      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 -3.6999999999999999e-4 < eps < 8.2000000000000006e-9

   1. Initial program 26.5%

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

      1. tan-sum 27.5%

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

      2. div-inv 27.5%

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

      3. fma-neg 27.6%

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

   3. Applied egg-rr 27.6%

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

   4. Taylor expanded in x around inf 27.5%

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

      1. associate--l+ 55.2%

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

   6. Simplified 55.3%

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

      1. tan-quot 53.2%

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

      2. tan-quot 55.3%

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

      3. div-inv 55.3%

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

      4. tan-quot 55.3%

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

      5. tan-quot 55.3%

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

      6. *-commutative 55.3%

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

   8. Applied egg-rr 55.2%

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

      1. fma-udef 55.3%

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

      2. *-rgt-identity 55.3%

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

      3. distribute-rgt-neg-in 55.3%

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

      4. metadata-eval 55.3%

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

      5. distribute-lft-out 55.2%

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

   10. Simplified 55.2%

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

   11. Taylor expanded in eps around 0 99.8%

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

       1. *-commutative 99.8%

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

       2. *-lft-identity 99.8%

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

       3. times-frac 99.8%

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

       4. /-rgt-identity 99.8%

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

       5. unpow2 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00037 \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left({\sin x}^{2} \cdot \frac{\varepsilon \cdot \varepsilon}{{\cos x}^{2}} + \left(\left(\frac{\varepsilon \cdot \sin x}{\cos x} - {\varepsilon}^{3} \cdot \left(\frac{\sin x}{\cos x} \cdot -0.3333333333333333 - \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{3}} - \frac{\sin x}{\cos x} \cdot -0.3333333333333333}{\cos x} - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -1.9e-7)
     (fma t_0 (/ 1.0 t_1) (- (tan x)))
     (if (<= eps 8.2e-9)
       (fma
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (* eps eps)
        (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0)))
       (- (/ t_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 <= -1.9e-7) {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	} else if (eps <= 8.2e-9) {
		tmp = fma(((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))), (eps * eps), (eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0)));
	} else {
		tmp = (t_0 / t_1) - 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 <= -1.9e-7)
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	elseif (eps <= 8.2e-9)
		tmp = fma(Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))), Float64(eps * eps), Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return 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, -1.9e-7], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 8.2e-9], N[(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] * N[(eps * 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]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.90000000000000007e-7

   1. Initial program 50.9%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -1.90000000000000007e-7 < eps < 8.2000000000000006e-9

   1. Initial program 26.1%

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

      1. tan-sum 26.5%

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

      2. div-inv 26.5%

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

      3. fma-neg 26.5%

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

   3. Applied egg-rr 26.5%

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

   4. Taylor expanded in eps around 0 99.6%

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

      1. fma-def 99.6%

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

      2. unpow2 99.6%

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

   6. Simplified 99.6%

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

   if 8.2000000000000006e-9 < eps

   1. Initial program 48.6%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{t_2}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon}{\frac{\cos x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_2} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -2.05e-7)
     (fma t_1 (/ 1.0 t_2) (- (tan x)))
     (if (<= eps 2.75e-9)
       (+
        (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
        (* (tan x) (/ eps (/ (cos x) (sin x)))))
       (- (/ t_1 t_2) (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 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -2.05e-7) {
		tmp = fma(t_1, (1.0 / t_2), -tan(x));
	} else if (eps <= 2.75e-9) {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + (tan(x) * (eps / (cos(x) / sin(x))));
	} else {
		tmp = (t_1 / t_2) - 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(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -2.05e-7)
		tmp = fma(t_1, Float64(1.0 / t_2), Float64(-tan(x)));
	elseif (eps <= 2.75e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(tan(x) * Float64(eps / Float64(cos(x) / sin(x)))));
	else
		tmp = Float64(Float64(t_1 / t_2) - tan(x));
	end
	return 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[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.05e-7], N[(t$95$1 * N[(1.0 / t$95$2), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.75e-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[(N[Tan[x], $MachinePrecision] * N[(eps / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.05e-7

   1. Initial program 50.2%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -2.05e-7 < eps < 2.7499999999999998e-9

   1. Initial program 26.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in x around inf 27.1%

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

      1. associate--l+ 55.0%

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

   6. Simplified 55.0%

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

      1. tan-quot 53.0%

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

      2. tan-quot 55.0%

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

      3. div-inv 55.0%

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

      4. tan-quot 55.0%

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

      5. tan-quot 55.0%

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

      6. *-commutative 55.0%

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

   8. Applied egg-rr 54.9%

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

      1. fma-udef 55.0%

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

      2. *-rgt-identity 55.0%

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

      3. distribute-rgt-neg-in 55.0%

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

      4. metadata-eval 55.0%

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

      5. distribute-lft-out 54.9%

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

   10. Simplified 54.9%

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

   11. Taylor expanded in eps around 0 99.5%

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

       1. associate-/l* 99.5%

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

   13. Simplified 99.5%

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

   if 2.7499999999999998e-9 < eps

   1. Initial program 48.6%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon}{\frac{\cos x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{t_2}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_2} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -2.05e-7)
     (fma t_1 (/ 1.0 t_2) (- (tan x)))
     (if (<= eps 7.3e-9)
       (+
        (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
        (* (tan x) (/ (* eps (sin x)) (cos x))))
       (- (/ t_1 t_2) (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 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -2.05e-7) {
		tmp = fma(t_1, (1.0 / t_2), -tan(x));
	} else if (eps <= 7.3e-9) {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + (tan(x) * ((eps * sin(x)) / cos(x)));
	} else {
		tmp = (t_1 / t_2) - 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(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -2.05e-7)
		tmp = fma(t_1, Float64(1.0 / t_2), Float64(-tan(x)));
	elseif (eps <= 7.3e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(tan(x) * Float64(Float64(eps * sin(x)) / cos(x))));
	else
		tmp = Float64(Float64(t_1 / t_2) - tan(x));
	end
	return 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[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.05e-7], N[(t$95$1 * N[(1.0 / t$95$2), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 7.3e-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[(N[Tan[x], $MachinePrecision] * N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.05e-7

   1. Initial program 50.2%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -2.05e-7 < eps < 7.30000000000000002e-9

   1. Initial program 26.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in x around inf 27.1%

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

      1. associate--l+ 55.0%

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

   6. Simplified 55.0%

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

      1. tan-quot 53.0%

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

      2. tan-quot 55.0%

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

      3. div-inv 55.0%

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

      4. tan-quot 55.0%

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

      5. tan-quot 55.0%

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

      6. *-commutative 55.0%

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

   8. Applied egg-rr 54.9%

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

      1. fma-udef 55.0%

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

      2. *-rgt-identity 55.0%

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

      3. distribute-rgt-neg-in 55.0%

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

      4. metadata-eval 55.0%

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

      5. distribute-lft-out 54.9%

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

   10. Simplified 54.9%

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

   11. Taylor expanded in eps around 0 99.5%

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

   if 7.30000000000000002e-9 < eps

   1. Initial program 48.6%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -2.3e-9)
     (fma t_0 (/ 1.0 t_1) (- (tan x)))
     (if (<= eps 3.25e-9)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (/ t_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 <= -2.3e-9) {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	} else if (eps <= 3.25e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 / t_1) - 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 <= -2.3e-9)
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	elseif (eps <= 3.25e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return 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, -2.3e-9], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 3.25e-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 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.2999999999999999e-9

   1. Initial program 50.9%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -2.2999999999999999e-9 < eps < 3.2500000000000002e-9

   1. Initial program 26.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

   if 3.2500000000000002e-9 < eps

   1. Initial program 48.6%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.1e-9) (not (<= eps 3.9e-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 <= -3.1e-9) || !(eps <= 3.9e-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 <= (-3.1d-9)) .or. (.not. (eps <= 3.9d-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 <= -3.1e-9) || !(eps <= 3.9e-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 <= -3.1e-9) or not (eps <= 3.9e-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 <= -3.1e-9) || !(eps <= 3.9e-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 <= -3.1e-9) || ~((eps <= 3.9e-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, -3.1e-9], N[Not[LessEqual[eps, 3.9e-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 < -3.10000000000000005e-9 or 3.9000000000000002e-9 < eps

   1. Initial program 49.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. fma-neg 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. fma-neg 99.2%

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

      2. associate-*r/ 99.2%

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

      3. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   if -3.10000000000000005e-9 < eps < 3.9000000000000002e-9

   1. Initial program 26.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00075 \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon - x \cdot \sin \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00075) (not (<= eps 8.2e-9)))
   (- (* (sin (+ eps x)) (/ 1.0 (- (cos eps) (* x (sin 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 <= -0.00075) || !(eps <= 8.2e-9)) {
		tmp = (sin((eps + x)) * (1.0 / (cos(eps) - (x * sin(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 <= (-0.00075d0)) .or. (.not. (eps <= 8.2d-9))) then
        tmp = (sin((eps + x)) * (1.0d0 / (cos(eps) - (x * sin(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 <= -0.00075) || !(eps <= 8.2e-9)) {
		tmp = (Math.sin((eps + x)) * (1.0 / (Math.cos(eps) - (x * Math.sin(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 <= -0.00075) or not (eps <= 8.2e-9):
		tmp = (math.sin((eps + x)) * (1.0 / (math.cos(eps) - (x * math.sin(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 <= -0.00075) || !(eps <= 8.2e-9))
		tmp = Float64(Float64(sin(Float64(eps + x)) * Float64(1.0 / Float64(cos(eps) - Float64(x * sin(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 <= -0.00075) || ~((eps <= 8.2e-9)))
		tmp = (sin((eps + x)) * (1.0 / (cos(eps) - (x * sin(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, -0.00075], N[Not[LessEqual[eps, 8.2e-9]], $MachinePrecision]], N[(N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[Cos[eps], $MachinePrecision] - N[(x * N[Sin[eps], $MachinePrecision]), $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 < -7.5000000000000002e-4 or 8.2000000000000006e-9 < eps

   1. Initial program 49.8%

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

      1. tan-quot 49.9%

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

      2. div-inv 49.7%

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

   3. Applied egg-rr 49.7%

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

   4. Taylor expanded in x around 0 53.4%

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

      1. mul-1-neg 53.4%

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

      2. unsub-neg 53.4%

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

      3. *-commutative 53.4%

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

   6. Simplified 53.4%

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

   if -7.5000000000000002e-4 < eps < 8.2000000000000006e-9

   1. Initial program 26.5%

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

   2. Taylor expanded in eps around 0 98.8%

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

      1. cancel-sign-sub-inv 98.8%

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

      2. metadata-eval 98.8%

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

      3. *-lft-identity 98.8%

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

      4. distribute-lft-in 98.8%

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

      5. *-rgt-identity 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00075 \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon - x \cdot \sin \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 8: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.9e-5)
   (fma (+ (tan x) (tan eps)) 1.0 (- (tan x)))
   (if (<= eps 8.2e-9)
     (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
     (- (* (sin (+ eps x)) (/ 1.0 (cos eps))) (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.9e-5) {
		tmp = fma((tan(x) + tan(eps)), 1.0, -tan(x));
	} else if (eps <= 8.2e-9) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = (sin((eps + x)) * (1.0 / cos(eps))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.9e-5)
		tmp = fma(Float64(tan(x) + tan(eps)), 1.0, Float64(-tan(x)));
	elseif (eps <= 8.2e-9)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(sin(Float64(eps + x)) * Float64(1.0 / cos(eps))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.9e-5], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * 1.0 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 8.2e-9], 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[(N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.9e-5

   1. Initial program 50.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.1%

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

      3. fma-neg 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around 0 53.3%

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

   if -4.9e-5 < eps < 8.2000000000000006e-9

   1. Initial program 26.5%

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

      1. tan-sum 27.5%

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

      2. div-inv 27.5%

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

      3. fma-neg 27.6%

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

   3. Applied egg-rr 27.6%

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

   4. Taylor expanded in eps around 0 98.8%

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

   if 8.2000000000000006e-9 < eps

   1. Initial program 48.6%

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

      1. tan-quot 48.6%

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

      2. div-inv 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in x around 0 51.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 9: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.6e-5)
   (fma (+ (tan x) (tan eps)) 1.0 (- (tan x)))
   (if (<= eps 8.2e-9)
     (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (- (* (sin (+ eps x)) (/ 1.0 (cos eps))) (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.6e-5) {
		tmp = fma((tan(x) + tan(eps)), 1.0, -tan(x));
	} else if (eps <= 8.2e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (sin((eps + x)) * (1.0 / cos(eps))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2.6e-5)
		tmp = fma(Float64(tan(x) + tan(eps)), 1.0, Float64(-tan(x)));
	elseif (eps <= 8.2e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(sin(Float64(eps + x)) * Float64(1.0 / cos(eps))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.6e-5], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * 1.0 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 8.2e-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[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.59999999999999984e-5

   1. Initial program 50.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.1%

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

      3. fma-neg 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around 0 53.3%

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

   if -2.59999999999999984e-5 < eps < 8.2000000000000006e-9

   1. Initial program 26.5%

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

   2. Taylor expanded in eps around 0 98.8%

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

      1. cancel-sign-sub-inv 98.8%

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

      2. metadata-eval 98.8%

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

      3. *-lft-identity 98.8%

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

      4. distribute-lft-in 98.8%

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

      5. *-rgt-identity 98.8%

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

   4. Simplified 98.8%

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

   if 8.2000000000000006e-9 < eps

   1. Initial program 48.6%

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

      1. tan-quot 48.6%

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

      2. div-inv 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in x around 0 51.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 10: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

2. Taylor expanded in x around 0 53.1%

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

3. Final simplification 53.1%

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

Alternative 11: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -160 \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -160.0) (not (<= eps 8.2e-9)))
   (- (tan (+ eps x)) x)
   (sin eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -160.0) || !(eps <= 8.2e-9)) {
		tmp = tan((eps + x)) - x;
	} else {
		tmp = sin(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 <= (-160.0d0)) .or. (.not. (eps <= 8.2d-9))) then
        tmp = tan((eps + x)) - x
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -160.0) || !(eps <= 8.2e-9)) {
		tmp = Math.tan((eps + x)) - x;
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -160.0) or not (eps <= 8.2e-9):
		tmp = math.tan((eps + x)) - x
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -160.0) || !(eps <= 8.2e-9))
		tmp = Float64(tan(Float64(eps + x)) - x);
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -160.0) || ~((eps <= 8.2e-9)))
		tmp = tan((eps + x)) - x;
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -160.0], N[Not[LessEqual[eps, 8.2e-9]], $MachinePrecision]], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision], N[Sin[eps], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -160 or 8.2000000000000006e-9 < eps

   1. Initial program 50.1%

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

   2. Taylor expanded in x around 0 46.9%

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

   if -160 < eps < 8.2000000000000006e-9

   1. Initial program 26.3%

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

      1. tan-quot 26.2%

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

      2. div-inv 25.9%

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

   3. Applied egg-rr 25.9%

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

   4. Taylor expanded in eps around 0 26.3%

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

      1. associate-*r* 26.3%

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

      2. mul-1-neg 26.3%

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

   6. Simplified 26.3%

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

   7. Taylor expanded in x around 0 54.0%

      \[\leadsto \color{blue}{\sin \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -160 \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 12: 35.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. tan-quot 38.8%

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

   2. div-inv 38.6%

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

3. Applied egg-rr 38.6%

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

4. Taylor expanded in eps around 0 19.9%

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

   1. associate-*r* 19.9%

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

   2. mul-1-neg 19.9%

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

6. Simplified 19.9%

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

7. Taylor expanded in x around 0 32.1%

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

8. Final simplification 32.1%

   \[\leadsto \sin \varepsilon \]

Alternative 13: 3.6% accurate, 102.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := (-x)

Derivation

1. Initial program 38.8%

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

2. Taylor expanded in x around 0 36.3%

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

3. Taylor expanded in x around inf 3.7%

   \[\leadsto \color{blue}{-1 \cdot x} \]
4. Step-by-step derivation

   1. neg-mul-1 3.7%

      \[\leadsto \color{blue}{-x} \]

5. Simplified 3.7%

   \[\leadsto \color{blue}{-x} \]

6. Final simplification 3.7%

   \[\leadsto -x \]

Developer target: 76.2% 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 2023187 
(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)))