2tan (problem 3.3.2)

Percentage Accurate: 42.4% → 98.9%

Time: 21.0s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ t_2 := \frac{\tan \varepsilon}{t_1}\\ \mathbf{if}\;\varepsilon \leq -3650000000 \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;t_2 + \mathsf{fma}\left(\frac{\tan x}{\cos x}, \frac{\cos x}{t_1}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{0.13333333333333333 \cdot \frac{{\varepsilon}^{5} \cdot t_0}{\cos x} + \left(0.3333333333333333 \cdot \frac{t_0 \cdot {\varepsilon}^{3}}{\cos x} + \frac{\varepsilon \cdot t_0}{\cos x}\right)}{t_1 \cdot \cos x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- 1.0 (* (tan eps) (tan x))))
        (t_2 (/ (tan eps) t_1)))
   (if (or (<= eps -3650000000.0) (not (<= eps 7e-14)))
     (+ t_2 (fma (/ (tan x) (cos x)) (/ (cos x) t_1) (- (tan x))))
     (+
      t_2
      (/
       (+
        (* 0.13333333333333333 (/ (* (pow eps 5.0) t_0) (cos x)))
        (+
         (* 0.3333333333333333 (/ (* t_0 (pow eps 3.0)) (cos x)))
         (/ (* eps t_0) (cos x))))
       (* t_1 (cos x)))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double t_2 = tan(eps) / t_1;
	double tmp;
	if ((eps <= -3650000000.0) || !(eps <= 7e-14)) {
		tmp = t_2 + fma((tan(x) / cos(x)), (cos(x) / t_1), -tan(x));
	} else {
		tmp = t_2 + (((0.13333333333333333 * ((pow(eps, 5.0) * t_0) / cos(x))) + ((0.3333333333333333 * ((t_0 * pow(eps, 3.0)) / cos(x))) + ((eps * t_0) / cos(x)))) / (t_1 * cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_2 = Float64(tan(eps) / t_1)
	tmp = 0.0
	if ((eps <= -3650000000.0) || !(eps <= 7e-14))
		tmp = Float64(t_2 + fma(Float64(tan(x) / cos(x)), Float64(cos(x) / t_1), Float64(-tan(x))));
	else
		tmp = Float64(t_2 + Float64(Float64(Float64(0.13333333333333333 * Float64(Float64((eps ^ 5.0) * t_0) / cos(x))) + Float64(Float64(0.3333333333333333 * Float64(Float64(t_0 * (eps ^ 3.0)) / cos(x))) + Float64(Float64(eps * t_0) / cos(x)))) / Float64(t_1 * cos(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[eps], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[eps, -3650000000.0], N[Not[LessEqual[eps, 7e-14]], $MachinePrecision]], N[(t$95$2 + N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(N[(0.13333333333333333 * N[(N[(N[Power[eps, 5.0], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(N[(t$95$0 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.65e9 or 7.0000000000000005e-14 < eps

   1. Initial program 48.7%

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

      1. tan-sum 99.4%

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

      2. flip-- 99.4%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 99.4%

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

      2. associate-/l* 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. unpow2 99.4%

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

      5. swap-sqr 99.4%

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

      6. metadata-eval 99.4%

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

      7. fma-udef 99.4%

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

      8. +-commutative 99.4%

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

      9. flip-- 99.4%

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

      10. associate-*l/ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. sub-neg 99.4%

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

      2. associate-+r- 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.5%

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

   9. Simplified 99.5%

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

       1. tan-quot 99.4%

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

       2. frac-sub 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. div-sub 99.3%

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

       2. sub-neg 99.3%

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

       3. *-commutative 99.3%

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

       4. times-frac 99.4%

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

       5. *-commutative 99.4%

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

       6. *-commutative 99.4%

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

       7. times-frac 99.4%

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

       8. tan-quot 99.5%

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

   13. Applied egg-rr 99.5%

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

       1. fma-def 99.5%

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

       2. distribute-lft-neg-in 99.5%

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

       3. *-inverses 99.5%

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

       4. *-rgt-identity 99.5%

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

   15. Simplified 99.5%

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

   if -3.65e9 < eps < 7.0000000000000005e-14

   1. Initial program 25.2%

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

      1. tan-sum 27.2%

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

      2. flip-- 27.2%

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

      3. associate-/r/ 27.2%

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

      4. metadata-eval 27.2%

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

   3. Applied egg-rr 27.2%

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

      1. associate-*l/ 27.2%

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

      2. +-commutative 27.2%

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

      3. fma-def 27.2%

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

      4. swap-sqr 27.2%

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

      5. unpow2 27.2%

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

   5. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 27.2%

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

      2. associate-/l* 27.2%

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

      3. *-un-lft-identity 27.2%

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

      4. unpow2 27.2%

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

      5. swap-sqr 27.2%

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

      6. metadata-eval 27.2%

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

      7. fma-udef 27.2%

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

      8. +-commutative 27.2%

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

      9. flip-- 27.2%

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

      10. associate-*l/ 27.2%

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

   7. Applied egg-rr 27.2%

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

      1. sub-neg 27.2%

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

      2. associate-+r- 27.2%

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

      3. +-commutative 27.2%

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

      4. associate--l+ 64.7%

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

   9. Simplified 64.7%

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

       1. tan-quot 63.3%

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

       2. frac-sub 63.1%

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

   11. Applied egg-rr 63.1%

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

   12. Taylor expanded in eps around 0 99.7%

       \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \frac{\color{blue}{0.13333333333333333 \cdot \frac{{\varepsilon}^{5} \cdot {\sin x}^{2}}{\cos x} + \left(0.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{\cos x} + \frac{\varepsilon \cdot {\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3650000000 \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \mathsf{fma}\left(\frac{\tan x}{\cos x}, \frac{\cos x}{1 - \tan \varepsilon \cdot \tan x}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \frac{0.13333333333333333 \cdot \frac{{\varepsilon}^{5} \cdot {\sin x}^{2}}{\cos x} + \left(0.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{\cos x} + \frac{\varepsilon \cdot {\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;t_1 + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \mathsf{fma}\left(\frac{\tan x}{\cos x}, \frac{\cos x}{t_0}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (/ (tan eps) t_0)))
   (if (<= eps -2.1e-6)
     (- (/ 1.0 (/ t_0 (+ (tan eps) (tan x)))) (tan x))
     (if (<= eps 7e-14)
       (+
        t_1
        (+
         (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
         (/ (pow eps 2.0) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))))
       (+ t_1 (fma (/ (tan x) (cos x)) (/ (cos x) t_0) (- (tan x))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) / t_0;
	double tmp;
	if (eps <= -2.1e-6) {
		tmp = (1.0 / (t_0 / (tan(eps) + tan(x)))) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = t_1 + (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + (pow(eps, 2.0) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))));
	} else {
		tmp = t_1 + fma((tan(x) / cos(x)), (cos(x) / t_0), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) / t_0)
	tmp = 0.0
	if (eps <= -2.1e-6)
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(tan(eps) + tan(x)))) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(t_1 + Float64(Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)) + Float64((eps ^ 2.0) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0)))));
	else
		tmp = Float64(t_1 + fma(Float64(tan(x) / cos(x)), Float64(cos(x) / t_0), Float64(-tan(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[eps, -2.1e-6], N[(N[(1.0 / N[(t$95$0 / N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(t$95$1 + N[(N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] / t$95$0), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.0999999999999998e-6

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -2.0999999999999998e-6 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

      1. tan-sum 23.7%

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

      2. flip-- 23.7%

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

      3. associate-/r/ 23.7%

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

      4. metadata-eval 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. associate-*l/ 23.7%

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

      2. +-commutative 23.7%

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

      3. fma-def 23.7%

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

      4. swap-sqr 23.7%

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

      5. unpow2 23.7%

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

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 23.7%

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

      2. associate-/l* 23.7%

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

      3. *-un-lft-identity 23.7%

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

      4. unpow2 23.7%

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

      5. swap-sqr 23.7%

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

      6. metadata-eval 23.7%

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

      7. fma-udef 23.7%

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

      8. +-commutative 23.7%

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

      9. flip-- 23.7%

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

      10. associate-*l/ 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. sub-neg 23.7%

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

      2. associate-+r- 23.7%

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

      3. +-commutative 23.7%

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

      4. associate--l+ 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-/l* 99.8%

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

   12. Simplified 99.8%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. flip-- 99.3%

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

      3. associate-/r/ 99.3%

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

      4. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. unpow2 99.3%

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

      5. swap-sqr 99.3%

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

      6. metadata-eval 99.3%

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

      7. fma-udef 99.3%

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

      8. +-commutative 99.3%

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

      9. flip-- 99.3%

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

      10. associate-*l/ 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-+r- 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.4%

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

   9. Simplified 99.4%

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

       1. tan-quot 99.3%

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

       2. frac-sub 99.1%

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

   11. Applied egg-rr 99.1%

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

       1. div-sub 99.2%

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

       2. sub-neg 99.2%

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

       3. *-commutative 99.2%

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

       4. times-frac 99.2%

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

       5. *-commutative 99.2%

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

       6. *-commutative 99.2%

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

       7. times-frac 99.3%

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

       8. tan-quot 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. fma-def 99.4%

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

       2. distribute-lft-neg-in 99.4%

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

       3. *-inverses 99.4%

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

       4. *-rgt-identity 99.4%

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

   15. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \mathsf{fma}\left(\frac{\tan x}{\cos x}, \frac{\cos x}{1 - \tan \varepsilon \cdot \tan x}, -\tan x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -7.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;t_1 + \frac{\frac{\varepsilon \cdot {\sin x}^{2}}{\cos x}}{t_0 \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \mathsf{fma}\left(\frac{\tan x}{\cos x}, \frac{\cos x}{t_0}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (/ (tan eps) t_0)))
   (if (<= eps -7.1e-6)
     (- (/ 1.0 (/ t_0 (+ (tan eps) (tan x)))) (tan x))
     (if (<= eps 7e-14)
       (+ t_1 (/ (/ (* eps (pow (sin x) 2.0)) (cos x)) (* t_0 (cos x))))
       (+ t_1 (fma (/ (tan x) (cos x)) (/ (cos x) t_0) (- (tan x))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) / t_0;
	double tmp;
	if (eps <= -7.1e-6) {
		tmp = (1.0 / (t_0 / (tan(eps) + tan(x)))) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = t_1 + (((eps * pow(sin(x), 2.0)) / cos(x)) / (t_0 * cos(x)));
	} else {
		tmp = t_1 + fma((tan(x) / cos(x)), (cos(x) / t_0), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) / t_0)
	tmp = 0.0
	if (eps <= -7.1e-6)
		tmp = Float64(Float64(1.0 / Float64(t_0 / Float64(tan(eps) + tan(x)))) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(t_1 + Float64(Float64(Float64(eps * (sin(x) ^ 2.0)) / cos(x)) / Float64(t_0 * cos(x))));
	else
		tmp = Float64(t_1 + fma(Float64(tan(x) / cos(x)), Float64(cos(x) / t_0), Float64(-tan(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[eps, -7.1e-6], N[(N[(1.0 / N[(t$95$0 / N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(t$95$1 + N[(N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] / t$95$0), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.0999999999999998e-6

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -7.0999999999999998e-6 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

      1. tan-sum 23.7%

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

      2. flip-- 23.7%

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

      3. associate-/r/ 23.7%

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

      4. metadata-eval 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. associate-*l/ 23.7%

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

      2. +-commutative 23.7%

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

      3. fma-def 23.7%

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

      4. swap-sqr 23.7%

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

      5. unpow2 23.7%

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

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 23.7%

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

      2. associate-/l* 23.7%

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

      3. *-un-lft-identity 23.7%

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

      4. unpow2 23.7%

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

      5. swap-sqr 23.7%

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

      6. metadata-eval 23.7%

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

      7. fma-udef 23.7%

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

      8. +-commutative 23.7%

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

      9. flip-- 23.7%

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

      10. associate-*l/ 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. sub-neg 23.7%

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

      2. associate-+r- 23.7%

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

      3. +-commutative 23.7%

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

      4. associate--l+ 63.1%

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

   9. Simplified 63.1%

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

       1. tan-quot 61.7%

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

       2. frac-sub 61.4%

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

   11. Applied egg-rr 61.4%

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

   12. Taylor expanded in eps around 0 99.7%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. flip-- 99.3%

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

      3. associate-/r/ 99.3%

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

      4. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. unpow2 99.3%

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

      5. swap-sqr 99.3%

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

      6. metadata-eval 99.3%

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

      7. fma-udef 99.3%

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

      8. +-commutative 99.3%

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

      9. flip-- 99.3%

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

      10. associate-*l/ 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-+r- 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.4%

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

   9. Simplified 99.4%

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

       1. tan-quot 99.3%

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

       2. frac-sub 99.1%

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

   11. Applied egg-rr 99.1%

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

       1. div-sub 99.2%

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

       2. sub-neg 99.2%

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

       3. *-commutative 99.2%

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

       4. times-frac 99.2%

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

       5. *-commutative 99.2%

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

       6. *-commutative 99.2%

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

       7. times-frac 99.3%

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

       8. tan-quot 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. fma-def 99.4%

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

       2. distribute-lft-neg-in 99.4%

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

       3. *-inverses 99.4%

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

       4. *-rgt-identity 99.4%

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

   15. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \frac{\frac{\varepsilon \cdot {\sin x}^{2}}{\cos x}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \mathsf{fma}\left(\frac{\tan x}{\cos x}, \frac{\cos x}{1 - \tan \varepsilon \cdot \tan x}, -\tan x\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t_0}{t_1}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \frac{{\sin x}^{2} \cdot \frac{\varepsilon}{\cos x}}{t_0 \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (+ (tan eps) (tan x))))
   (if (<= eps -5.5e-6)
     (- (/ 1.0 (/ t_0 t_1)) (tan x))
     (if (<= eps 7e-14)
       (+
        (/ (tan eps) t_0)
        (/ (* (pow (sin x) 2.0) (/ eps (cos x))) (* t_0 (cos x))))
       (-
        (* t_1 (/ 1.0 (- 1.0 (* (/ (tan x) (cos eps)) (sin eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -5.5e-6) {
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = (tan(eps) / t_0) + ((pow(sin(x), 2.0) * (eps / cos(x))) / (t_0 * cos(x)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(eps) * tan(x))
    t_1 = tan(eps) + tan(x)
    if (eps <= (-5.5d-6)) then
        tmp = (1.0d0 / (t_0 / t_1)) - tan(x)
    else if (eps <= 7d-14) then
        tmp = (tan(eps) / t_0) + (((sin(x) ** 2.0d0) * (eps / cos(x))) / (t_0 * cos(x)))
    else
        tmp = (t_1 * (1.0d0 / (1.0d0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double t_1 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -5.5e-6) {
		tmp = (1.0 / (t_0 / t_1)) - Math.tan(x);
	} else if (eps <= 7e-14) {
		tmp = (Math.tan(eps) / t_0) + ((Math.pow(Math.sin(x), 2.0) * (eps / Math.cos(x))) / (t_0 * Math.cos(x)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((Math.tan(x) / Math.cos(eps)) * Math.sin(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(eps) * math.tan(x))
	t_1 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -5.5e-6:
		tmp = (1.0 / (t_0 / t_1)) - math.tan(x)
	elif eps <= 7e-14:
		tmp = (math.tan(eps) / t_0) + ((math.pow(math.sin(x), 2.0) * (eps / math.cos(x))) / (t_0 * math.cos(x)))
	else:
		tmp = (t_1 * (1.0 / (1.0 - ((math.tan(x) / math.cos(eps)) * math.sin(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -5.5e-6)
		tmp = Float64(Float64(1.0 / Float64(t_0 / t_1)) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(Float64(tan(eps) / t_0) + Float64(Float64((sin(x) ^ 2.0) * Float64(eps / cos(x))) / Float64(t_0 * cos(x))));
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) / cos(eps)) * sin(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(eps) * tan(x));
	t_1 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -5.5e-6)
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	elseif (eps <= 7e-14)
		tmp = (tan(eps) / t_0) + (((sin(x) ^ 2.0) * (eps / cos(x))) / (t_0 * cos(x)));
	else
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.5e-6], N[(N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.4999999999999999e-6

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -5.4999999999999999e-6 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

      1. tan-sum 23.7%

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

      2. flip-- 23.7%

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

      3. associate-/r/ 23.7%

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

      4. metadata-eval 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. associate-*l/ 23.7%

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

      2. +-commutative 23.7%

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

      3. fma-def 23.7%

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

      4. swap-sqr 23.7%

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

      5. unpow2 23.7%

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

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 23.7%

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

      2. associate-/l* 23.7%

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

      3. *-un-lft-identity 23.7%

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

      4. unpow2 23.7%

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

      5. swap-sqr 23.7%

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

      6. metadata-eval 23.7%

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

      7. fma-udef 23.7%

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

      8. +-commutative 23.7%

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

      9. flip-- 23.7%

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

      10. associate-*l/ 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. sub-neg 23.7%

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

      2. associate-+r- 23.7%

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

      3. +-commutative 23.7%

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

      4. associate--l+ 63.1%

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

   9. Simplified 63.1%

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

       1. tan-quot 61.7%

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

       2. frac-sub 61.4%

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

   11. Applied egg-rr 61.4%

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

   12. Taylor expanded in eps around 0 99.7%

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

       1. associate-/l* 99.7%

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

       2. associate-/r/ 99.7%

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

   14. Simplified 99.7%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. tan-quot 99.3%

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

      2. associate-*r/ 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \frac{{\sin x}^{2} \cdot \frac{\varepsilon}{\cos x}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t_0}{t_1}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \frac{\frac{\varepsilon \cdot {\sin x}^{2}}{\cos x}}{t_0 \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (+ (tan eps) (tan x))))
   (if (<= eps -4.8e-6)
     (- (/ 1.0 (/ t_0 t_1)) (tan x))
     (if (<= eps 7e-14)
       (+
        (/ (tan eps) t_0)
        (/ (/ (* eps (pow (sin x) 2.0)) (cos x)) (* t_0 (cos x))))
       (-
        (* t_1 (/ 1.0 (- 1.0 (* (/ (tan x) (cos eps)) (sin eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -4.8e-6) {
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = (tan(eps) / t_0) + (((eps * pow(sin(x), 2.0)) / cos(x)) / (t_0 * cos(x)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(eps) * tan(x))
    t_1 = tan(eps) + tan(x)
    if (eps <= (-4.8d-6)) then
        tmp = (1.0d0 / (t_0 / t_1)) - tan(x)
    else if (eps <= 7d-14) then
        tmp = (tan(eps) / t_0) + (((eps * (sin(x) ** 2.0d0)) / cos(x)) / (t_0 * cos(x)))
    else
        tmp = (t_1 * (1.0d0 / (1.0d0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double t_1 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -4.8e-6) {
		tmp = (1.0 / (t_0 / t_1)) - Math.tan(x);
	} else if (eps <= 7e-14) {
		tmp = (Math.tan(eps) / t_0) + (((eps * Math.pow(Math.sin(x), 2.0)) / Math.cos(x)) / (t_0 * Math.cos(x)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((Math.tan(x) / Math.cos(eps)) * Math.sin(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(eps) * math.tan(x))
	t_1 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -4.8e-6:
		tmp = (1.0 / (t_0 / t_1)) - math.tan(x)
	elif eps <= 7e-14:
		tmp = (math.tan(eps) / t_0) + (((eps * math.pow(math.sin(x), 2.0)) / math.cos(x)) / (t_0 * math.cos(x)))
	else:
		tmp = (t_1 * (1.0 / (1.0 - ((math.tan(x) / math.cos(eps)) * math.sin(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -4.8e-6)
		tmp = Float64(Float64(1.0 / Float64(t_0 / t_1)) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(Float64(tan(eps) / t_0) + Float64(Float64(Float64(eps * (sin(x) ^ 2.0)) / cos(x)) / Float64(t_0 * cos(x))));
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) / cos(eps)) * sin(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(eps) * tan(x));
	t_1 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -4.8e-6)
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	elseif (eps <= 7e-14)
		tmp = (tan(eps) / t_0) + (((eps * (sin(x) ^ 2.0)) / cos(x)) / (t_0 * cos(x)));
	else
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.8e-6], N[(N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.7999999999999998e-6

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -4.7999999999999998e-6 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

      1. tan-sum 23.7%

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

      2. flip-- 23.7%

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

      3. associate-/r/ 23.7%

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

      4. metadata-eval 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. associate-*l/ 23.7%

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

      2. +-commutative 23.7%

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

      3. fma-def 23.7%

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

      4. swap-sqr 23.7%

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

      5. unpow2 23.7%

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

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 23.7%

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

      2. associate-/l* 23.7%

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

      3. *-un-lft-identity 23.7%

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

      4. unpow2 23.7%

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

      5. swap-sqr 23.7%

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

      6. metadata-eval 23.7%

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

      7. fma-udef 23.7%

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

      8. +-commutative 23.7%

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

      9. flip-- 23.7%

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

      10. associate-*l/ 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. sub-neg 23.7%

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

      2. associate-+r- 23.7%

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

      3. +-commutative 23.7%

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

      4. associate--l+ 63.1%

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

   9. Simplified 63.1%

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

       1. tan-quot 61.7%

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

       2. frac-sub 61.4%

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

   11. Applied egg-rr 61.4%

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

   12. Taylor expanded in eps around 0 99.7%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. tan-quot 99.3%

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

      2. associate-*r/ 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \frac{\frac{\varepsilon \cdot {\sin x}^{2}}{\cos x}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{t_0}{t_1}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (+ (tan eps) (tan x))))
   (if (<= eps -7.1e-9)
     (- (/ 1.0 (/ t_0 t_1)) (tan x))
     (if (<= eps 7e-14)
       (+ (/ (tan eps) t_0) (* (pow (sin x) 2.0) (/ eps (pow (cos x) 2.0))))
       (-
        (* t_1 (/ 1.0 (- 1.0 (* (/ (tan x) (cos eps)) (sin eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -7.1e-9) {
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = (tan(eps) / t_0) + (pow(sin(x), 2.0) * (eps / pow(cos(x), 2.0)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(eps) * tan(x))
    t_1 = tan(eps) + tan(x)
    if (eps <= (-7.1d-9)) then
        tmp = (1.0d0 / (t_0 / t_1)) - tan(x)
    else if (eps <= 7d-14) then
        tmp = (tan(eps) / t_0) + ((sin(x) ** 2.0d0) * (eps / (cos(x) ** 2.0d0)))
    else
        tmp = (t_1 * (1.0d0 / (1.0d0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double t_1 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -7.1e-9) {
		tmp = (1.0 / (t_0 / t_1)) - Math.tan(x);
	} else if (eps <= 7e-14) {
		tmp = (Math.tan(eps) / t_0) + (Math.pow(Math.sin(x), 2.0) * (eps / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_1 * (1.0 / (1.0 - ((Math.tan(x) / Math.cos(eps)) * Math.sin(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(eps) * math.tan(x))
	t_1 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -7.1e-9:
		tmp = (1.0 / (t_0 / t_1)) - math.tan(x)
	elif eps <= 7e-14:
		tmp = (math.tan(eps) / t_0) + (math.pow(math.sin(x), 2.0) * (eps / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_1 * (1.0 / (1.0 - ((math.tan(x) / math.cos(eps)) * math.sin(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -7.1e-9)
		tmp = Float64(Float64(1.0 / Float64(t_0 / t_1)) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(Float64(tan(eps) / t_0) + Float64((sin(x) ^ 2.0) * Float64(eps / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) / cos(eps)) * sin(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(eps) * tan(x));
	t_1 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -7.1e-9)
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	elseif (eps <= 7e-14)
		tmp = (tan(eps) / t_0) + ((sin(x) ^ 2.0) * (eps / (cos(x) ^ 2.0)));
	else
		tmp = (t_1 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7.1e-9], N[(N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(eps / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.09999999999999988e-9

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -7.09999999999999988e-9 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

      1. tan-sum 23.7%

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

      2. flip-- 23.7%

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

      3. associate-/r/ 23.7%

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

      4. metadata-eval 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. associate-*l/ 23.7%

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

      2. +-commutative 23.7%

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

      3. fma-def 23.7%

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

      4. swap-sqr 23.7%

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

      5. unpow2 23.7%

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

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - {\tan x}^{2} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} - \tan x \]
   6. Step-by-step derivation

      1. sub-neg 23.7%

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

      2. associate-/l* 23.7%

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

      3. *-un-lft-identity 23.7%

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

      4. unpow2 23.7%

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

      5. swap-sqr 23.7%

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

      6. metadata-eval 23.7%

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

      7. fma-udef 23.7%

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

      8. +-commutative 23.7%

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

      9. flip-- 23.7%

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

      10. associate-*l/ 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. sub-neg 23.7%

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

      2. associate-+r- 23.7%

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

      3. +-commutative 23.7%

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

      4. associate--l+ 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in eps around 0 99.5%

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

       1. associate-/l* 99.4%

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

       2. associate-/r/ 99.5%

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

   12. Simplified 99.5%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. tan-quot 99.3%

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

      2. associate-*r/ 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \]

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -6.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{t_0}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -6.1e-9)
     (- (/ 1.0 (/ (- 1.0 (* (tan eps) (tan x))) t_0)) (tan x))
     (if (<= eps 7e-14)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (-
        (* t_0 (/ 1.0 (- 1.0 (* (/ (tan x) (cos eps)) (sin eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -6.1e-9) {
		tmp = (1.0 / ((1.0 - (tan(eps) * tan(x))) / t_0)) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(eps) + tan(x)
    if (eps <= (-6.1d-9)) then
        tmp = (1.0d0 / ((1.0d0 - (tan(eps) * tan(x))) / t_0)) - tan(x)
    else if (eps <= 7d-14) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 * (1.0d0 / (1.0d0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -6.1e-9) {
		tmp = (1.0 / ((1.0 - (Math.tan(eps) * Math.tan(x))) / t_0)) - Math.tan(x);
	} else if (eps <= 7e-14) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - ((Math.tan(x) / Math.cos(eps)) * Math.sin(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -6.1e-9:
		tmp = (1.0 / ((1.0 - (math.tan(eps) * math.tan(x))) / t_0)) - math.tan(x)
	elif eps <= 7e-14:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 * (1.0 / (1.0 - ((math.tan(x) / math.cos(eps)) * math.sin(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -6.1e-9)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 - Float64(tan(eps) * tan(x))) / t_0)) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) / cos(eps)) * sin(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -6.1e-9)
		tmp = (1.0 / ((1.0 - (tan(eps) * tan(x))) / t_0)) - tan(x);
	elseif (eps <= 7e-14)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 * (1.0 / (1.0 - ((tan(x) / cos(eps)) * sin(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.1e-9], N[(N[(1.0 / N[(N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.1e-9

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -6.1e-9 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. mul-1-neg 99.4%

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

      3. remove-double-neg 99.4%

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

   4. Simplified 99.4%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. tan-quot 99.3%

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

      2. associate-*r/ 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon} - \tan x\\ \end{array} \]

Alternative 8: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -6.2e-9) (not (<= eps 7e-14)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.2e-9) || !(eps <= 7e-14)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	} else {
		tmp = eps * (1.0 + (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 <= (-6.2d-9)) .or. (.not. (eps <= 7d-14))) then
        tmp = ((tan(eps) + tan(x)) / (1.0d0 - (tan(eps) * tan(x)))) - tan(x)
    else
        tmp = eps * (1.0d0 + ((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 <= -6.2e-9) || !(eps <= 7e-14)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / (1.0 - (Math.tan(eps) * Math.tan(x)))) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (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 <= -6.2e-9) or not (eps <= 7e-14):
		tmp = ((math.tan(eps) + math.tan(x)) / (1.0 - (math.tan(eps) * math.tan(x)))) - math.tan(x)
	else:
		tmp = eps * (1.0 + (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 <= -6.2e-9) || !(eps <= 7e-14))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + 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 <= -6.2e-9) || ~((eps <= 7e-14)))
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -6.2e-9], N[Not[LessEqual[eps, 7e-14]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(1.0 + 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 < -6.2000000000000001e-9 or 7.0000000000000005e-14 < eps

   1. Initial program 49.5%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

   if -6.2000000000000001e-9 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. mul-1-neg 99.4%

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

      3. remove-double-neg 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 9: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{t_0}{t_1}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (+ (tan eps) (tan x))))
   (if (<= eps -6.4e-9)
     (- (/ 1.0 (/ t_0 t_1)) (tan x))
     (if (<= eps 7e-14)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (/ t_1 t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -6.4e-9) {
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	} else if (eps <= 7e-14) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_1 / t_0) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(eps) * tan(x))
    t_1 = tan(eps) + tan(x)
    if (eps <= (-6.4d-9)) then
        tmp = (1.0d0 / (t_0 / t_1)) - tan(x)
    else if (eps <= 7d-14) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_1 / t_0) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double t_1 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -6.4e-9) {
		tmp = (1.0 / (t_0 / t_1)) - Math.tan(x);
	} else if (eps <= 7e-14) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_1 / t_0) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(eps) * math.tan(x))
	t_1 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -6.4e-9:
		tmp = (1.0 / (t_0 / t_1)) - math.tan(x)
	elif eps <= 7e-14:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_1 / t_0) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -6.4e-9)
		tmp = Float64(Float64(1.0 / Float64(t_0 / t_1)) - tan(x));
	elseif (eps <= 7e-14)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_1 / t_0) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(eps) * tan(x));
	t_1 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -6.4e-9)
		tmp = (1.0 / (t_0 / t_1)) - tan(x);
	elseif (eps <= 7e-14)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_1 / t_0) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.4e-9], N[(N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7e-14], N[(eps * N[(1.0 + 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$1 / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.40000000000000023e-9

   1. Initial program 49.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. tan-quot 98.9%

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

      3. div-inv 98.9%

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

      4. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. associate-/l* 99.2%

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

      3. fma-udef 99.2%

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

      4. +-commutative 99.2%

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

      5. div-inv 99.2%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -6.40000000000000023e-9 < eps < 7.0000000000000005e-14

   1. Initial program 23.1%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. mul-1-neg 99.4%

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

      3. remove-double-neg 99.4%

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

   4. Simplified 99.4%

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

   if 7.0000000000000005e-14 < eps

   1. Initial program 49.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 10: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0037 \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0037) (not (<= eps 7e-14)))
   (tan eps)
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0037) || !(eps <= 7e-14)) {
		tmp = tan(eps);
	} else {
		tmp = eps * (1.0 + (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 <= (-0.0037d0)) .or. (.not. (eps <= 7d-14))) then
        tmp = tan(eps)
    else
        tmp = eps * (1.0d0 + ((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 <= -0.0037) || !(eps <= 7e-14)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * (1.0 + (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 <= -0.0037) or not (eps <= 7e-14):
		tmp = math.tan(eps)
	else:
		tmp = eps * (1.0 + (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 <= -0.0037) || !(eps <= 7e-14))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(1.0 + 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 <= -0.0037) || ~((eps <= 7e-14)))
		tmp = tan(eps);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0037], N[Not[LessEqual[eps, 7e-14]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(1.0 + 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 < -0.0037000000000000002 or 7.0000000000000005e-14 < eps

   1. Initial program 50.2%

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

   2. Taylor expanded in x around 0 53.1%

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

      1. expm1-log1p-u 41.3%

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

      2. expm1-udef 40.3%

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

      3. tan-quot 40.5%

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

   4. Applied egg-rr 40.5%

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

      1. expm1-def 41.5%

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

      2. expm1-log1p 53.4%

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

   6. Simplified 53.4%

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

   if -0.0037000000000000002 < eps < 7.0000000000000005e-14

   1. Initial program 22.7%

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

   2. Taylor expanded in eps around 0 98.2%

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

      1. sub-neg 98.2%

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

      2. mul-1-neg 98.2%

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

      3. remove-double-neg 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0037 \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 11: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

2. Taylor expanded in x around 0 57.3%

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

   1. expm1-log1p-u 51.1%

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

   2. expm1-udef 24.5%

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

   3. tan-quot 24.6%

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

4. Applied egg-rr 24.6%

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

   1. expm1-def 51.2%

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

   2. expm1-log1p 57.4%

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

6. Simplified 57.4%

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

7. Final simplification 57.4%

   \[\leadsto \tan \varepsilon \]

Alternative 12: 30.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 37.1%

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

2. Taylor expanded in x around 0 57.3%

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

3. Taylor expanded in eps around 0 32.0%

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

4. Final simplification 32.0%

   \[\leadsto \varepsilon \]

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 2023305 
(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)))