2tan (problem 3.3.2)

Percentage Accurate: 42.0% → 99.4%

Time: 19.8s

Alternatives: 17

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: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - t_1\\ t_3 := \tan x + \tan \varepsilon\\ t_4 := {\sin x}^{2}\\ t_5 := \frac{t_4}{t_0}\\ t_6 := 1 + t_5\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_3}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;{\varepsilon}^{3} \cdot \left(\left(\frac{t_4 \cdot t_6}{t_0} + \left(-0.5 \cdot \left(-1 - t_5\right) - 0.16666666666666666 \cdot t_5\right)\right) - 0.16666666666666666\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot t_6\right)}{\cos x} + \varepsilon \cdot t_6\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_2, -\sin x, t_3 \cdot \cos x\right) + \mathsf{fma}\left(\sin x, -1 + t_1, \sin x \cdot t_2\right)}{\cos x \cdot t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (* (tan x) (tan eps)))
        (t_2 (- 1.0 t_1))
        (t_3 (+ (tan x) (tan eps)))
        (t_4 (pow (sin x) 2.0))
        (t_5 (/ t_4 t_0))
        (t_6 (+ 1.0 t_5)))
   (if (<= eps -4.8e-5)
     (- (/ t_3 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 3.3e-6)
       (+
        (*
         (pow eps 3.0)
         (-
          (+
           (/ (* t_4 t_6) t_0)
           (- (* -0.5 (- -1.0 t_5)) (* 0.16666666666666666 t_5)))
          0.16666666666666666))
        (+ (/ (* (pow eps 2.0) (* (sin x) t_6)) (cos x)) (* eps t_6)))
       (/
        (+
         (fma t_2 (- (sin x)) (* t_3 (cos x)))
         (fma (sin x) (+ -1.0 t_1) (* (sin x) t_2)))
        (* (cos x) t_2))))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = tan(x) * tan(eps);
	double t_2 = 1.0 - t_1;
	double t_3 = tan(x) + tan(eps);
	double t_4 = pow(sin(x), 2.0);
	double t_5 = t_4 / t_0;
	double t_6 = 1.0 + t_5;
	double tmp;
	if (eps <= -4.8e-5) {
		tmp = (t_3 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 3.3e-6) {
		tmp = (pow(eps, 3.0) * ((((t_4 * t_6) / t_0) + ((-0.5 * (-1.0 - t_5)) - (0.16666666666666666 * t_5))) - 0.16666666666666666)) + (((pow(eps, 2.0) * (sin(x) * t_6)) / cos(x)) + (eps * t_6));
	} else {
		tmp = (fma(t_2, -sin(x), (t_3 * cos(x))) + fma(sin(x), (-1.0 + t_1), (sin(x) * t_2))) / (cos(x) * t_2);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64(tan(x) * tan(eps))
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(tan(x) + tan(eps))
	t_4 = sin(x) ^ 2.0
	t_5 = Float64(t_4 / t_0)
	t_6 = Float64(1.0 + t_5)
	tmp = 0.0
	if (eps <= -4.8e-5)
		tmp = Float64(Float64(t_3 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 3.3e-6)
		tmp = Float64(Float64((eps ^ 3.0) * Float64(Float64(Float64(Float64(t_4 * t_6) / t_0) + Float64(Float64(-0.5 * Float64(-1.0 - t_5)) - Float64(0.16666666666666666 * t_5))) - 0.16666666666666666)) + Float64(Float64(Float64((eps ^ 2.0) * Float64(sin(x) * t_6)) / cos(x)) + Float64(eps * t_6)));
	else
		tmp = Float64(Float64(fma(t_2, Float64(-sin(x)), Float64(t_3 * cos(x))) + fma(sin(x), Float64(-1.0 + t_1), Float64(sin(x) * t_2))) / Float64(cos(x) * t_2));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[(1.0 + t$95$5), $MachinePrecision]}, If[LessEqual[eps, -4.8e-5], N[(N[(t$95$3 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.3e-6], N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[(N[(t$95$4 * t$95$6), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$5), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * (-N[Sin[x], $MachinePrecision]) + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.8000000000000001e-5

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. tan-quot 99.5%

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

       3. associate-*r/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -4.8000000000000001e-5 < eps < 3.30000000000000017e-6

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 99.6%

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

   if 3.30000000000000017e-6 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. tan-quot 99.3%

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

      3. frac-sub 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. prod-diff 99.5%

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

      2. *-commutative 99.5%

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

      3. fma-neg 99.4%

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

      4. *-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. fma-def 99.4%

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

      6. *-commutative 99.4%

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

      7. fma-udef 99.3%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;{\varepsilon}^{3} \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - 0.16666666666666666\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \tan x \cdot \tan \varepsilon, -\sin x, \left(\tan x + \tan \varepsilon\right) \cdot \cos x\right) + \mathsf{fma}\left(\sin x, -1 + \tan x \cdot \tan \varepsilon, \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - t_1\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_2, -\sin x, t_0 \cdot \cos x\right) + \mathsf{fma}\left(\sin x, -1 + t_1, \sin x \cdot t_2\right)}{\cos x \cdot t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps)))
        (t_1 (* (tan x) (tan eps)))
        (t_2 (- 1.0 t_1)))
   (if (<= eps -3.6e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 6e-7)
       (+
        (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (/
        (+
         (fma t_2 (- (sin x)) (* t_0 (cos x)))
         (fma (sin x) (+ -1.0 t_1) (* (sin x) t_2)))
        (* (cos x) t_2))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = tan(x) * tan(eps);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (eps <= -3.6e-7) {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 6e-7) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (fma(t_2, -sin(x), (t_0 * cos(x))) + fma(sin(x), (-1.0 + t_1), (sin(x) * t_2))) / (cos(x) * t_2);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(tan(x) * tan(eps))
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (eps <= -3.6e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 6e-7)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(fma(t_2, Float64(-sin(x)), Float64(t_0 * cos(x))) + fma(sin(x), Float64(-1.0 + t_1), Float64(sin(x) * t_2))) / Float64(cos(x) * t_2));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[eps, -3.6e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6e-7], N[(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[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * (-N[Sin[x], $MachinePrecision]) + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.59999999999999994e-7

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. tan-quot 99.5%

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

       3. associate-*r/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -3.59999999999999994e-7 < eps < 5.9999999999999997e-7

   1. Initial program 29.6%

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

      1. tan-sum 30.8%

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

      2. div-inv 30.8%

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

      3. *-un-lft-identity 30.8%

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

      4. prod-diff 30.8%

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

      5. *-commutative 30.8%

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

      6. *-un-lft-identity 30.8%

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

      7. *-commutative 30.8%

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

      8. *-un-lft-identity 30.8%

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

   3. Applied egg-rr 30.8%

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

      1. +-commutative 30.8%

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

      2. fma-udef 30.8%

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

      3. associate-+r+ 30.8%

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

      4. unsub-neg 30.8%

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

   5. Simplified 30.8%

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

      1. tan-quot 30.8%

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

      2. clear-num 30.8%

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

      3. un-div-inv 30.8%

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

      4. clear-num 30.8%

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

      5. tan-quot 30.8%

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

   7. Applied egg-rr 30.8%

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

   8. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-lft-identity 99.6%

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

   10. Simplified 99.6%

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

   if 5.9999999999999997e-7 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. tan-quot 99.3%

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

      3. frac-sub 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. prod-diff 99.5%

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

      2. *-commutative 99.5%

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

      3. fma-neg 99.4%

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

      4. *-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. fma-def 99.4%

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

      6. *-commutative 99.4%

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

      7. fma-udef 99.3%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \tan x \cdot \tan \varepsilon, -\sin x, \left(\tan x + \tan \varepsilon\right) \cdot \cos x\right) + \mathsf{fma}\left(\sin x, -1 + \tan x \cdot \tan \varepsilon, \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\sin x}^{2} \cdot {\cos x}^{-2}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -1.35e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 6e-7)
       (+
        (* eps (+ 1.0 (* (pow (sin x) 2.0) (pow (cos x) -2.0))))
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -1.35e-7) {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 6e-7) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) * pow(cos(x), -2.0)))) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-1.35d-7)) then
        tmp = (t_0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x)
    else if (eps <= 6d-7) then
        tmp = (eps * (1.0d0 + ((sin(x) ** 2.0d0) * (cos(x) ** (-2.0d0))))) + ((eps ** 2.0d0) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -1.35e-7) {
		tmp = (t_0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 6e-7) {
		tmp = (eps * (1.0 + (Math.pow(Math.sin(x), 2.0) * Math.pow(Math.cos(x), -2.0)))) + (Math.pow(eps, 2.0) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -1.35e-7:
		tmp = (t_0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps)))) - math.tan(x)
	elif eps <= 6e-7:
		tmp = (eps * (1.0 + (math.pow(math.sin(x), 2.0) * math.pow(math.cos(x), -2.0)))) + (math.pow(eps, 2.0) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -1.35e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 6e-7)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) * (cos(x) ^ -2.0)))) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -1.35e-7)
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	elseif (eps <= 6e-7)
		tmp = (eps * (1.0 + ((sin(x) ^ 2.0) * (cos(x) ^ -2.0)))) + ((eps ^ 2.0) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.35e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6e-7], N[(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[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.35000000000000004e-7

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. tan-quot 99.5%

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

       3. associate-*r/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -1.35000000000000004e-7 < eps < 5.9999999999999997e-7

   1. Initial program 29.6%

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

      1. tan-sum 30.8%

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

      2. div-inv 30.8%

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

      3. *-un-lft-identity 30.8%

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

      4. prod-diff 30.8%

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

      5. *-commutative 30.8%

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

      6. *-un-lft-identity 30.8%

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

      7. *-commutative 30.8%

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

      8. *-un-lft-identity 30.8%

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

   3. Applied egg-rr 30.8%

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

      1. +-commutative 30.8%

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

      2. fma-udef 30.8%

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

      3. associate-+r+ 30.8%

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

      4. unsub-neg 30.8%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

   8. Simplified 99.5%

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

   if 5.9999999999999997e-7 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\sin x}^{2} \cdot {\cos x}^{-2}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2.7e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 3.9e-7)
       (+
        (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.7e-7) {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 3.9e-7) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-2.7d-7)) then
        tmp = (t_0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x)
    else if (eps <= 3.9d-7) then
        tmp = (eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps ** 2.0d0) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -2.7e-7) {
		tmp = (t_0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 3.9e-7) {
		tmp = (eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + (Math.pow(eps, 2.0) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -2.7e-7:
		tmp = (t_0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps)))) - math.tan(x)
	elif eps <= 3.9e-7:
		tmp = (eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + (math.pow(eps, 2.0) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.7e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 3.9e-7)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -2.7e-7)
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	elseif (eps <= 3.9e-7)
		tmp = (eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps ^ 2.0) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.7e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.9e-7], N[(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[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.70000000000000009e-7

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. tan-quot 99.5%

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

       3. associate-*r/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -2.70000000000000009e-7 < eps < 3.90000000000000025e-7

   1. Initial program 29.6%

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

      1. tan-sum 30.8%

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

      2. div-inv 30.8%

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

      3. *-un-lft-identity 30.8%

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

      4. prod-diff 30.8%

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

      5. *-commutative 30.8%

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

      6. *-un-lft-identity 30.8%

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

      7. *-commutative 30.8%

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

      8. *-un-lft-identity 30.8%

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

   3. Applied egg-rr 30.8%

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

      1. +-commutative 30.8%

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

      2. fma-udef 30.8%

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

      3. associate-+r+ 30.8%

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

      4. unsub-neg 30.8%

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

   5. Simplified 30.8%

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

      1. tan-quot 30.8%

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

      2. clear-num 30.8%

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

      3. un-div-inv 30.8%

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

      4. clear-num 30.8%

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

      5. tan-quot 30.8%

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

   7. Applied egg-rr 30.8%

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

   8. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-lft-identity 99.6%

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

   10. Simplified 99.6%

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

   if 3.90000000000000025e-7 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -8.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -8.9e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 4.4e-7)
       (/ (* eps (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))) (* (cos x) t_1))
       (- (/ t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -8.9e-7) {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 4.4e-7) {
		tmp = (eps * (cos(x) + (pow(sin(x), 2.0) / cos(x)))) / (cos(x) * t_1);
	} else {
		tmp = (t_0 / t_1) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    if (eps <= (-8.9d-7)) then
        tmp = (t_0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x)
    else if (eps <= 4.4d-7) then
        tmp = (eps * (cos(x) + ((sin(x) ** 2.0d0) / cos(x)))) / (cos(x) * t_1)
    else
        tmp = (t_0 / t_1) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double tmp;
	if (eps <= -8.9e-7) {
		tmp = (t_0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 4.4e-7) {
		tmp = (eps * (Math.cos(x) + (Math.pow(Math.sin(x), 2.0) / Math.cos(x)))) / (Math.cos(x) * t_1);
	} else {
		tmp = (t_0 / t_1) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	tmp = 0
	if eps <= -8.9e-7:
		tmp = (t_0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps)))) - math.tan(x)
	elif eps <= 4.4e-7:
		tmp = (eps * (math.cos(x) + (math.pow(math.sin(x), 2.0) / math.cos(x)))) / (math.cos(x) * t_1)
	else:
		tmp = (t_0 / t_1) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -8.9e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 4.4e-7)
		tmp = Float64(Float64(eps * Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x)))) / Float64(cos(x) * t_1));
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = 1.0 - (tan(x) * tan(eps));
	tmp = 0.0;
	if (eps <= -8.9e-7)
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	elseif (eps <= 4.4e-7)
		tmp = (eps * (cos(x) + ((sin(x) ^ 2.0) / cos(x)))) / (cos(x) * t_1);
	else
		tmp = (t_0 / t_1) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -8.9e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.4e-7], N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -8.899999999999999e-7

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. tan-quot 99.5%

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

       3. associate-*r/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -8.899999999999999e-7 < eps < 4.4000000000000002e-7

   1. Initial program 29.6%

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

      1. tan-sum 30.8%

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

      2. tan-quot 30.8%

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

      3. frac-sub 30.7%

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

   3. Applied egg-rr 30.7%

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

   4. Taylor expanded in eps around 0 99.5%

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

      1. sub-neg 99.5%

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

      2. mul-1-neg 99.5%

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

      3. remove-double-neg 99.5%

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

   6. Simplified 99.5%

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

   if 4.4000000000000002e-7 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \left(-1 + \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -1.95e-9)
     (- (/ t_0 (- 1.0 (+ -1.0 (fma (tan x) (tan eps) 1.0)))) (tan x))
     (if (<= eps 2.6e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -1.95e-9) {
		tmp = (t_0 / (1.0 - (-1.0 + fma(tan(x), tan(eps), 1.0)))) - tan(x);
	} else if (eps <= 2.6e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -1.95e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(-1.0 + fma(tan(x), tan(eps), 1.0)))) - tan(x));
	elseif (eps <= 2.6e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.95e-9], N[(N[(t$95$0 / N[(1.0 - N[(-1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.6e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.9500000000000001e-9

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. expm1-log1p-u 81.2%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

      2. expm1-udef 81.2%

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

      3. log1p-udef 81.1%

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

      4. add-exp-log 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.6%

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

      4. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

   if -1.9500000000000001e-9 < eps < 2.6000000000000001e-9

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

   4. Simplified 99.2%

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

      1. distribute-rgt-in 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. frac-times 99.3%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.3%

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

      8. pow2 99.3%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.3%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.4%

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

   8. Simplified 99.4%

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

   if 2.6000000000000001e-9 < eps

   1. Initial program 51.9%

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

      1. tan-sum 98.7%

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

      2. div-inv 98.5%

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

      3. *-un-lft-identity 98.5%

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

      4. prod-diff 98.6%

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

      5. *-commutative 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. *-un-lft-identity 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-udef 98.5%

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

      3. associate-+r+ 98.5%

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

      4. unsub-neg 98.5%

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

   5. Simplified 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(-1 + \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 7: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.7e-9)
     (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 4.4e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 4.4e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.7e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 4.4e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.7e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.4e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.7e-9

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. tan-quot 99.5%

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

       3. associate-*r/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -3.7e-9 < eps < 4.3999999999999997e-9

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

   4. Simplified 99.2%

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

      1. distribute-rgt-in 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. frac-times 99.3%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.3%

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

      8. pow2 99.3%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.3%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.4%

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

   8. Simplified 99.4%

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

   if 4.3999999999999997e-9 < eps

   1. Initial program 51.9%

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

      1. tan-sum 98.7%

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

      2. div-inv 98.5%

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

      3. *-un-lft-identity 98.5%

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

      4. prod-diff 98.6%

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

      5. *-commutative 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. *-un-lft-identity 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-udef 98.5%

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

      3. associate-+r+ 98.5%

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

      4. unsub-neg 98.5%

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

   5. Simplified 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 8: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.9e-9) (not (<= eps 3.3e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.9e-9) || !(eps <= 3.3e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.9e-9) || !(eps <= 3.3e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.9e-9], N[Not[LessEqual[eps, 3.3e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.89999999999999991e-9 or 3.30000000000000018e-9 < eps

   1. Initial program 53.7%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-udef 99.0%

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

      3. associate-+r+ 99.0%

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

      4. unsub-neg 99.0%

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

   5. Simplified 99.2%

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

   if -2.89999999999999991e-9 < eps < 3.30000000000000018e-9

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

   4. Simplified 99.2%

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

      1. distribute-rgt-in 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. frac-times 99.3%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.3%

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

      8. pow2 99.3%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.3%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.4%

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

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 9: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.6e-9)
     (- (/ t_0 (- 1.0 (/ (tan eps) (/ 1.0 (tan x))))) (tan x))
     (if (<= eps 2.8e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = (t_0 / (1.0 - (tan(eps) / (1.0 / tan(x))))) - tan(x);
	} else if (eps <= 2.8e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.6e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(eps) / Float64(1.0 / tan(x))))) - tan(x));
	elseif (eps <= 2.8e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.6e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.8e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5999999999999998e-9

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

       1. clear-num 99.5%

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

       2. associate-*l/ 99.5%

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

       3. *-un-lft-identity 99.5%

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

       4. clear-num 99.6%

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

       5. tan-quot 99.6%

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

   11. Applied egg-rr 99.6%

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

   if -4.5999999999999998e-9 < eps < 2.79999999999999984e-9

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

   4. Simplified 99.2%

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

      1. distribute-rgt-in 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. unpow2 99.3%

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

      4. unpow2 99.3%

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

      5. frac-times 99.3%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.3%

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

      8. pow2 99.3%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.3%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.4%

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

   8. Simplified 99.4%

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

   if 2.79999999999999984e-9 < eps

   1. Initial program 51.9%

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

      1. tan-sum 98.7%

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

      2. div-inv 98.5%

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

      3. *-un-lft-identity 98.5%

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

      4. prod-diff 98.6%

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

      5. *-commutative 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. *-un-lft-identity 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-udef 98.5%

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

      3. associate-+r+ 98.5%

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

      4. unsub-neg 98.5%

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

   5. Simplified 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 10: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.5e-5) (not (<= eps 1.75e-6)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* x (tan eps)))) (tan x))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.5e-5) || !(eps <= 1.75e-6)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (x * tan(eps)))) - tan(x);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.5e-5) || !(eps <= 1.75e-6))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(x * tan(eps)))) - tan(x));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.5e-5], N[Not[LessEqual[eps, 1.75e-6]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.500000000000001e-5 or 1.74999999999999997e-6 < eps

   1. Initial program 54.1%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.4%

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

      3. clear-num 99.4%

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

      4. tan-quot 99.4%

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

      5. frac-times 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. clear-num 99.3%

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

      8. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. times-frac 99.4%

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

      4. remove-double-div 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around 0 57.8%

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

   if -8.500000000000001e-5 < eps < 1.74999999999999997e-6

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

   4. Simplified 98.9%

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

      1. distribute-rgt-in 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. unpow2 99.0%

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

      4. unpow2 99.0%

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

      5. frac-times 98.9%

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

      6. tan-quot 99.0%

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

      7. tan-quot 99.0%

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

      8. pow2 99.0%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.0%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.0%

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

   8. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 11: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -5e-6)
     (- (/ t_0 (- 1.0 (* x (tan eps)))) (tan x))
     (if (<= eps 1.4e-6)
       (fma eps (pow (tan x) 2.0) eps)
       (/ 1.0 (/ (- 1.0 (* (tan x) (tan eps))) t_0))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -5e-6) {
		tmp = (t_0 / (1.0 - (x * tan(eps)))) - tan(x);
	} else if (eps <= 1.4e-6) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = 1.0 / ((1.0 - (tan(x) * tan(eps))) / t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -5e-6)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(x * tan(eps)))) - tan(x));
	elseif (eps <= 1.4e-6)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 - Float64(tan(x) * tan(eps))) / t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5e-6], N[(N[(t$95$0 / N[(1.0 - N[(x * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.4e-6], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(1.0 / N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.00000000000000041e-6

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in x around 0 59.2%

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

   if -5.00000000000000041e-6 < eps < 1.39999999999999994e-6

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

   4. Simplified 98.9%

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

      1. distribute-rgt-in 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. unpow2 99.0%

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

      4. unpow2 99.0%

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

      5. frac-times 98.9%

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

      6. tan-quot 99.0%

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

      7. tan-quot 99.0%

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

      8. pow2 99.0%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.0%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.0%

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

   8. Simplified 99.0%

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

   if 1.39999999999999994e-6 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.4%

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

      1. tan-sum 52.8%

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

      2. add-sqr-sqrt 30.3%

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

      3. sqrt-unprod 34.6%

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

      4. pow2 34.6%

         \[\leadsto \sqrt{\color{blue}{{\tan \left(x + \varepsilon\right)}^{2}}} - \tan x \]

   7. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{\sqrt{{\tan \left(x + \varepsilon\right)}^{2}}} - \tan x \]
   8. Step-by-step derivation

      1. unpow2 34.6%

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

      2. rem-sqrt-square 34.6%

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

      3. +-commutative 34.6%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in x around 0 34.4%

       \[\leadsto \color{blue}{\left|\tan \left(\varepsilon + x\right)\right|} \]
   11. Step-by-step derivation

       1. +-commutative 34.4%

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

       2. rem-square-sqrt 30.8%

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

       3. fabs-sqr 30.8%

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

       4. rem-square-sqrt 54.0%

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

       5. +-commutative 54.0%

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

   12. Simplified 54.0%

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

       1. +-commutative 54.0%

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

       2. tan-sum 56.1%

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

       3. clear-num 55.9%

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

   14. Applied egg-rr 55.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}\\ \end{array} \]

Alternative 12: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.1e-5)
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* x (tan eps)))) (tan x))
   (if (<= eps 3.3e-6)
     (fma eps (pow (tan x) 2.0) eps)
     (/ (- (- (tan x)) (tan eps)) (+ -1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.1e-5) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (x * tan(eps)))) - tan(x);
	} else if (eps <= 3.3e-6) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (-tan(x) - tan(eps)) / (-1.0 + (tan(x) * tan(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.1e-5)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(x * tan(eps)))) - tan(x));
	elseif (eps <= 3.3e-6)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(Float64(-tan(x)) - tan(eps)) / Float64(-1.0 + Float64(tan(x) * tan(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.1e-5], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.3e-6], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.1e-5

   1. Initial program 55.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.5%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.4%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. remove-double-div 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in x around 0 59.2%

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

   if -1.1e-5 < eps < 3.30000000000000017e-6

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

   4. Simplified 98.9%

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

      1. distribute-rgt-in 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. unpow2 99.0%

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

      4. unpow2 99.0%

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

      5. frac-times 98.9%

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

      6. tan-quot 99.0%

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

      7. tan-quot 99.0%

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

      8. pow2 99.0%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.0%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.0%

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

   8. Simplified 99.0%

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

   if 3.30000000000000017e-6 < eps

   1. Initial program 52.8%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   5. Simplified 99.4%

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

      1. tan-sum 52.8%

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

      2. add-sqr-sqrt 30.3%

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

      3. sqrt-unprod 34.6%

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

      4. pow2 34.6%

         \[\leadsto \sqrt{\color{blue}{{\tan \left(x + \varepsilon\right)}^{2}}} - \tan x \]

   7. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{\sqrt{{\tan \left(x + \varepsilon\right)}^{2}}} - \tan x \]
   8. Step-by-step derivation

      1. unpow2 34.6%

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

      2. rem-sqrt-square 34.6%

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

      3. +-commutative 34.6%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in x around 0 34.4%

       \[\leadsto \color{blue}{\left|\tan \left(\varepsilon + x\right)\right|} \]
   11. Step-by-step derivation

       1. +-commutative 34.4%

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

       2. rem-square-sqrt 30.8%

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

       3. fabs-sqr 30.8%

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

       4. rem-square-sqrt 54.0%

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

       5. +-commutative 54.0%

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

   12. Simplified 54.0%

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

       1. +-commutative 54.0%

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

       2. tan-sum 56.1%

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

       3. frac-2neg 56.1%

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

   14. Applied egg-rr 56.1%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}\\ \end{array} \]

Alternative 13: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.9e-6) (not (<= eps 1.8e-6)))
   (tan eps)
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.9e-6) || !(eps <= 1.8e-6)) {
		tmp = tan(eps);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.9e-6) || !(eps <= 1.8e-6))
		tmp = tan(eps);
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.9e-6], N[Not[LessEqual[eps, 1.8e-6]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.9e-6 or 1.79999999999999992e-6 < eps

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0 56.8%

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

      1. tan-quot 57.0%

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

      2. expm1-log1p-u 45.8%

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

      3. expm1-udef 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 57.0%

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

   6. Simplified 57.0%

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

   if -1.9e-6 < eps < 1.79999999999999992e-6

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

   4. Simplified 98.9%

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

      1. distribute-rgt-in 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. unpow2 99.0%

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

      4. unpow2 99.0%

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

      5. frac-times 98.9%

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

      6. tan-quot 99.0%

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

      7. tan-quot 99.0%

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

      8. pow2 99.0%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

      2. *-commutative 99.0%

         \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2}} + \varepsilon \]

      3. fma-def 99.0%

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

   8. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 14: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.65 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.65e-5) (not (<= eps 3.1e-6)))
   (tan eps)
   (* eps (+ 1.0 (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.65e-5) || !(eps <= 3.1e-6)) {
		tmp = tan(eps);
	} else {
		tmp = eps * (1.0 + pow(tan(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 <= (-1.65d-5)) .or. (.not. (eps <= 3.1d-6))) then
        tmp = tan(eps)
    else
        tmp = eps * (1.0d0 + (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.65e-5) || !(eps <= 3.1e-6)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * (1.0 + Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.65e-5) or not (eps <= 3.1e-6):
		tmp = math.tan(eps)
	else:
		tmp = eps * (1.0 + math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.65e-5) || !(eps <= 3.1e-6))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(1.0 + (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.65e-5) || ~((eps <= 3.1e-6)))
		tmp = tan(eps);
	else
		tmp = eps * (1.0 + (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.65e-5], N[Not[LessEqual[eps, 3.1e-6]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.6500000000000001e-5 or 3.1e-6 < eps

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0 56.8%

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

      1. tan-quot 57.0%

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

      2. expm1-log1p-u 45.8%

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

      3. expm1-udef 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. expm1-def 45.8%

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

      2. expm1-log1p 57.0%

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

   6. Simplified 57.0%

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

   if -1.6500000000000001e-5 < eps < 3.1e-6

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

   4. Simplified 98.9%

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

      1. unpow2 98.9%

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

      2. unpow2 98.9%

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

      3. frac-times 98.9%

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

      4. tan-quot 98.9%

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

      5. tan-quot 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

   8. Simplified 98.9%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.65 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \]

Alternative 15: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.3e-5) (not (<= eps 1e-6)))
   (tan eps)
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-5) || !(eps <= 1e-6)) {
		tmp = tan(eps);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.3d-5)) .or. (.not. (eps <= 1d-6))) then
        tmp = tan(eps)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-5) || !(eps <= 1e-6)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.3e-5) or not (eps <= 1e-6):
		tmp = math.tan(eps)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.3e-5) || !(eps <= 1e-6))
		tmp = tan(eps);
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.3e-5) || ~((eps <= 1e-6)))
		tmp = tan(eps);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.3e-5], N[Not[LessEqual[eps, 1e-6]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.3000000000000003e-5 or 9.99999999999999955e-7 < eps

   1. Initial program 54.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in x around 0 56.8%

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
   3. Step-by-step derivation

      1. tan-quot 57.0%

         \[\leadsto \color{blue}{\tan \varepsilon} \]

      2. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

      3. expm1-udef 45.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]

   4. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

      2. expm1-log1p 57.0%

         \[\leadsto \color{blue}{\tan \varepsilon} \]

   6. Simplified 57.0%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

   if -3.3000000000000003e-5 < eps < 9.99999999999999955e-7

   1. Initial program 29.6%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 98.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 98.9%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 98.9%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.0%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.0%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.0%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 98.9%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.0%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.0%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.0%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 16: 58.4% 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 42.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in x around 0 56.4%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
3. Step-by-step derivation

   1. tan-quot 56.4%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

   2. expm1-log1p-u 50.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

   3. expm1-udef 26.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]

4. Applied egg-rr 26.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-def 50.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

   2. expm1-log1p 56.4%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

6. Simplified 56.4%

   \[\leadsto \color{blue}{\tan \varepsilon} \]

7. Final simplification 56.4%

   \[\leadsto \tan \varepsilon \]

Alternative 17: 31.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 42.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in x around 0 56.4%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

3. Taylor expanded in eps around 0 29.1%

   \[\leadsto \color{blue}{\varepsilon} \]

4. Final simplification 29.1%

   \[\leadsto \varepsilon \]

Developer target: 76.4% 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 2023334 
(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)))