2tan (problem 3.3.2)

Percentage Accurate: 42.2% → 99.5%

Time: 18.9s

Alternatives: 14

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: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 - \tan \varepsilon \cdot \tan x}\\ t_1 := \frac{-1}{\cos x}\\ t_2 := \frac{\sin x}{\cos x}\\ t_3 := \frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot t_2}\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;t_3 + \tan x \cdot \left(t_0 + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(t_2 + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\mathsf{fma}\left(t_1, \sin x, \sin x \cdot \frac{1}{\cos x}\right) + \mathsf{fma}\left(\tan x, t_0, \sin x \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 1.0 (* (tan eps) (tan x)))))
        (t_1 (/ -1.0 (cos x)))
        (t_2 (/ (sin x) (cos x)))
        (t_3 (/ (tan eps) (- 1.0 (* (/ (sin eps) (cos eps)) t_2)))))
   (if (<= eps -5.8e-8)
     (+ t_3 (* (tan x) (+ t_0 -1.0)))
     (if (<= eps 2e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (* (* eps eps) (+ t_2 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (+
        t_3
        (+
         (fma t_1 (sin x) (* (sin x) (/ 1.0 (cos x))))
         (fma (tan x) t_0 (* (sin x) t_1))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / (1.0 - (tan(eps) * tan(x)));
	double t_1 = -1.0 / cos(x);
	double t_2 = sin(x) / cos(x);
	double t_3 = tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * t_2));
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = t_3 + (tan(x) * (t_0 + -1.0));
	} else if (eps <= 2e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * (t_2 + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = t_3 + (fma(t_1, sin(x), (sin(x) * (1.0 / cos(x)))) + fma(tan(x), t_0, (sin(x) * t_1)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 / Float64(1.0 - Float64(tan(eps) * tan(x))))
	t_1 = Float64(-1.0 / cos(x))
	t_2 = Float64(sin(x) / cos(x))
	t_3 = Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * t_2)))
	tmp = 0.0
	if (eps <= -5.8e-8)
		tmp = Float64(t_3 + Float64(tan(x) * Float64(t_0 + -1.0)));
	elseif (eps <= 2e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(t_2 + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(t_3 + Float64(fma(t_1, sin(x), Float64(sin(x) * Float64(1.0 / cos(x)))) + fma(tan(x), t_0, Float64(sin(x) * t_1))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.8e-8], N[(t$95$3 + N[(N[Tan[x], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(t$95$2 + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(t$95$1 * N[Sin[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * t$95$0 + N[(N[Sin[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.8000000000000003e-8

   1. Initial program 47.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.1%

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

      1. tan-quot 99.3%

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

      2. expm1-log1p-u 75.3%

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

      3. expm1-udef 74.8%

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

   10. Applied egg-rr 74.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 75.3%

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

       2. expm1-log1p 99.3%

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

   12. Simplified 99.3%

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

       1. tan-quot 99.3%

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

       2. tan-quot 99.4%

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

       3. div-inv 99.4%

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

       4. fma-neg 99.4%

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

       5. tan-quot 99.5%

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

       6. tan-quot 99.5%

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

       7. *-commutative 99.5%

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

       8. *-commutative 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. fma-udef 99.5%

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

       2. *-rgt-identity 99.5%

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

       3. distribute-rgt-neg-in 99.5%

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

       4. metadata-eval 99.5%

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

       5. distribute-lft-out 99.5%

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

       6. *-commutative 99.5%

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

   16. Simplified 99.5%

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

   if -5.8000000000000003e-8 < eps < 1.9999999999999999e-7

   1. Initial program 31.1%

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

      1. tan-sum 32.4%

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

      2. div-inv 32.4%

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

      3. fma-neg 32.4%

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

   3. Applied egg-rr 32.4%

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

      1. fma-neg 32.4%

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

      2. associate-*r/ 32.4%

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

      3. *-rgt-identity 32.4%

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

   5. Simplified 32.4%

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

   6. Taylor expanded in eps around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   8. Simplified 99.8%

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

   if 1.9999999999999999e-7 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf 99.2%

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

      1. associate--l+ 99.3%

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

   8. Simplified 99.3%

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

      1. tan-quot 99.6%

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

      2. expm1-log1p-u 69.4%

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

      3. expm1-udef 69.1%

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

   10. Applied egg-rr 69.1%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 69.4%

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

       2. expm1-log1p 99.6%

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

   12. Simplified 99.6%

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

       1. tan-quot 99.5%

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

       2. div-inv 99.5%

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

       3. div-inv 99.6%

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

       4. prod-diff 99.6%

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

   14. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

       2. distribute-neg-frac 99.7%

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

       3. metadata-eval 99.7%

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

       4. *-commutative 99.7%

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

       5. *-commutative 99.7%

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

       6. distribute-lft-neg-in 99.7%

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

       7. distribute-neg-frac 99.7%

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

       8. metadata-eval 99.7%

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

   16. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \sin x \cdot \frac{1}{\cos x}\right) + \mathsf{fma}\left(\tan x, \frac{1}{1 - \tan \varepsilon \cdot \tan x}, \sin x \cdot \frac{-1}{\cos x}\right)\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := 1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot t_0\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_1} + \tan x \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 3.05 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(t_0 + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \left(t_0 - \frac{t_0}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (- 1.0 (* (/ (sin eps) (cos eps)) t_0))))
   (if (<= eps -5.8e-8)
     (+
      (/ (tan eps) t_1)
      (* (tan x) (+ (/ 1.0 (- 1.0 (* (tan eps) (tan x)))) -1.0)))
     (if (<= eps 3.05e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (* (* eps eps) (+ t_0 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (-
        (/ (tan eps) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
        (- t_0 (/ t_0 t_1)))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = 1.0 - ((sin(eps) / cos(eps)) * t_0);
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = (tan(eps) / t_1) + (tan(x) * ((1.0 / (1.0 - (tan(eps) * tan(x)))) + -1.0));
	} else if (eps <= 3.05e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * (t_0 + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (tan(eps) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - (t_0 - (t_0 / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(x) / cos(x)
    t_1 = 1.0d0 - ((sin(eps) / cos(eps)) * t_0)
    if (eps <= (-5.8d-8)) then
        tmp = (tan(eps) / t_1) + (tan(x) * ((1.0d0 / (1.0d0 - (tan(eps) * tan(x)))) + (-1.0d0)))
    else if (eps <= 3.05d-7) then
        tmp = (eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps * eps) * (t_0 + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = (tan(eps) / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - (t_0 - (t_0 / t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) / Math.cos(x);
	double t_1 = 1.0 - ((Math.sin(eps) / Math.cos(eps)) * t_0);
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = (Math.tan(eps) / t_1) + (Math.tan(x) * ((1.0 / (1.0 - (Math.tan(eps) * Math.tan(x)))) + -1.0));
	} else if (eps <= 3.05e-7) {
		tmp = (eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + ((eps * eps) * (t_0 + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = (Math.tan(eps) / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - (t_0 - (t_0 / t_1));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(x) / math.cos(x)
	t_1 = 1.0 - ((math.sin(eps) / math.cos(eps)) * t_0)
	tmp = 0
	if eps <= -5.8e-8:
		tmp = (math.tan(eps) / t_1) + (math.tan(x) * ((1.0 / (1.0 - (math.tan(eps) * math.tan(x)))) + -1.0))
	elif eps <= 3.05e-7:
		tmp = (eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + ((eps * eps) * (t_0 + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = (math.tan(eps) / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - (t_0 - (t_0 / t_1))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * t_0))
	tmp = 0.0
	if (eps <= -5.8e-8)
		tmp = Float64(Float64(tan(eps) / t_1) + Float64(tan(x) * Float64(Float64(1.0 / Float64(1.0 - Float64(tan(eps) * tan(x)))) + -1.0)));
	elseif (eps <= 3.05e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(t_0 + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - Float64(t_0 - Float64(t_0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) / cos(x);
	t_1 = 1.0 - ((sin(eps) / cos(eps)) * t_0);
	tmp = 0.0;
	if (eps <= -5.8e-8)
		tmp = (tan(eps) / t_1) + (tan(x) * ((1.0 / (1.0 - (tan(eps) * tan(x)))) + -1.0));
	elseif (eps <= 3.05e-7)
		tmp = (eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps * eps) * (t_0 + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = (tan(eps) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - (t_0 - (t_0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.8e-8], N[(N[(N[Tan[eps], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(1.0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.05e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(t$95$0 + 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[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.8000000000000003e-8

   1. Initial program 47.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.1%

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

      1. tan-quot 99.3%

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

      2. expm1-log1p-u 75.3%

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

      3. expm1-udef 74.8%

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

   10. Applied egg-rr 74.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 75.3%

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

       2. expm1-log1p 99.3%

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

   12. Simplified 99.3%

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

       1. tan-quot 99.3%

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

       2. tan-quot 99.4%

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

       3. div-inv 99.4%

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

       4. fma-neg 99.4%

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

       5. tan-quot 99.5%

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

       6. tan-quot 99.5%

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

       7. *-commutative 99.5%

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

       8. *-commutative 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. fma-udef 99.5%

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

       2. *-rgt-identity 99.5%

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

       3. distribute-rgt-neg-in 99.5%

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

       4. metadata-eval 99.5%

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

       5. distribute-lft-out 99.5%

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

       6. *-commutative 99.5%

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

   16. Simplified 99.5%

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

   if -5.8000000000000003e-8 < eps < 3.04999999999999991e-7

   1. Initial program 31.1%

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

      1. tan-sum 32.4%

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

      2. div-inv 32.4%

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

      3. fma-neg 32.4%

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

   3. Applied egg-rr 32.4%

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

      1. fma-neg 32.4%

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

      2. associate-*r/ 32.4%

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

      3. *-rgt-identity 32.4%

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

   5. Simplified 32.4%

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

   6. Taylor expanded in eps around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   8. Simplified 99.8%

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

   if 3.04999999999999991e-7 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf 99.2%

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

      1. associate--l+ 99.3%

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

   8. Simplified 99.3%

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

      1. tan-quot 99.6%

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

      2. expm1-log1p-u 69.4%

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

      3. expm1-udef 69.1%

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

   10. Applied egg-rr 69.1%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 69.4%

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

       2. expm1-log1p 99.6%

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

   12. Simplified 99.6%

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

       1. tan-quot 99.6%

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

       2. associate-*r/ 99.6%

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

   14. Applied egg-rr 99.6%

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

       1. associate-/l* 99.6%

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

   16. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 3.05 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \left(\frac{\sin x}{\cos x} - \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot t_1} + \tan x \cdot \left(\frac{1}{t_0} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(t_1 + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (/ (sin x) (cos x))))
   (if (<= eps -5.8e-8)
     (+
      (/ (tan eps) (- 1.0 (* (/ (sin eps) (cos eps)) t_1)))
      (* (tan x) (+ (/ 1.0 t_0) -1.0)))
     (if (<= eps 3e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (* (* eps eps) (+ t_1 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- (/ (+ (tan eps) (tan x)) t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = sin(x) / cos(x);
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = (tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * t_1))) + (tan(x) * ((1.0 / t_0) + -1.0));
	} else if (eps <= 3e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * (t_1 + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = ((tan(eps) + tan(x)) / t_0) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(eps) * tan(x))
    t_1 = sin(x) / cos(x)
    if (eps <= (-5.8d-8)) then
        tmp = (tan(eps) / (1.0d0 - ((sin(eps) / cos(eps)) * t_1))) + (tan(x) * ((1.0d0 / t_0) + (-1.0d0)))
    else if (eps <= 3d-7) then
        tmp = (eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps * eps) * (t_1 + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = ((tan(eps) + tan(x)) / t_0) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double t_1 = Math.sin(x) / Math.cos(x);
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = (Math.tan(eps) / (1.0 - ((Math.sin(eps) / Math.cos(eps)) * t_1))) + (Math.tan(x) * ((1.0 / t_0) + -1.0));
	} else if (eps <= 3e-7) {
		tmp = (eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + ((eps * eps) * (t_1 + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = ((Math.tan(eps) + Math.tan(x)) / t_0) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(eps) * math.tan(x))
	t_1 = math.sin(x) / math.cos(x)
	tmp = 0
	if eps <= -5.8e-8:
		tmp = (math.tan(eps) / (1.0 - ((math.sin(eps) / math.cos(eps)) * t_1))) + (math.tan(x) * ((1.0 / t_0) + -1.0))
	elif eps <= 3e-7:
		tmp = (eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + ((eps * eps) * (t_1 + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = ((math.tan(eps) + math.tan(x)) / t_0) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -5.8e-8)
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * t_1))) + Float64(tan(x) * Float64(Float64(1.0 / t_0) + -1.0)));
	elseif (eps <= 3e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(t_1 + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / t_0) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(eps) * tan(x));
	t_1 = sin(x) / cos(x);
	tmp = 0.0;
	if (eps <= -5.8e-8)
		tmp = (tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * t_1))) + (tan(x) * ((1.0 / t_0) + -1.0));
	elseif (eps <= 3e-7)
		tmp = (eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps * eps) * (t_1 + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = ((tan(eps) + tan(x)) / t_0) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.8e-8], N[(N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(t$95$1 + 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[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.8000000000000003e-8

   1. Initial program 47.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.1%

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

      1. tan-quot 99.3%

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

      2. expm1-log1p-u 75.3%

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

      3. expm1-udef 74.8%

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

   10. Applied egg-rr 74.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 75.3%

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

       2. expm1-log1p 99.3%

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

   12. Simplified 99.3%

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

       1. tan-quot 99.3%

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

       2. tan-quot 99.4%

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

       3. div-inv 99.4%

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

       4. fma-neg 99.4%

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

       5. tan-quot 99.5%

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

       6. tan-quot 99.5%

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

       7. *-commutative 99.5%

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

       8. *-commutative 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. fma-udef 99.5%

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

       2. *-rgt-identity 99.5%

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

       3. distribute-rgt-neg-in 99.5%

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

       4. metadata-eval 99.5%

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

       5. distribute-lft-out 99.5%

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

       6. *-commutative 99.5%

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

   16. Simplified 99.5%

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

   if -5.8000000000000003e-8 < eps < 2.9999999999999999e-7

   1. Initial program 31.1%

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

      1. tan-sum 32.4%

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

      2. div-inv 32.4%

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

      3. fma-neg 32.4%

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

   3. Applied egg-rr 32.4%

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

      1. fma-neg 32.4%

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

      2. associate-*r/ 32.4%

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

      3. *-rgt-identity 32.4%

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

   5. Simplified 32.4%

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

   6. Taylor expanded in eps around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   8. Simplified 99.8%

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

   if 2.9999999999999999e-7 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -2.6e-9)
     (+
      (/ (tan eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x)))))
      (- (/ (tan x) t_0) (tan x)))
     (if (<= eps 4.6e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ (+ (tan eps) (tan x)) t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -2.6e-9) {
		tmp = (tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))))) + ((tan(x) / t_0) - tan(x));
	} else if (eps <= 4.6e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = ((tan(eps) + tan(x)) / t_0) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -2.6e-9)
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x))))) + Float64(Float64(tan(x) / t_0) - tan(x)));
	elseif (eps <= 4.6e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / t_0) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.6e-9], N[(N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.6e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.6000000000000001e-9

   1. Initial program 47.0%

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

      1. tan-sum 98.8%

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

      2. div-inv 98.8%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. fma-neg 98.8%

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

      2. associate-*r/ 98.8%

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

      3. *-rgt-identity 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 98.4%

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

      1. associate--l+ 98.4%

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

   8. Simplified 98.6%

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

      1. tan-quot 98.8%

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

      2. expm1-log1p-u 75.2%

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

      3. expm1-udef 74.6%

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

   10. Applied egg-rr 74.6%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 75.2%

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

       2. expm1-log1p 98.8%

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

   12. Simplified 98.8%

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

       1. tan-quot 98.8%

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

       2. sub-neg 98.8%

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

       3. tan-quot 98.9%

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

       4. tan-quot 99.0%

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

       5. tan-quot 99.0%

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

   14. Applied egg-rr 99.0%

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

       1. sub-neg 99.0%

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

       2. *-commutative 99.0%

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

   16. Simplified 99.0%

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

   if -2.6000000000000001e-9 < eps < 4.5999999999999998e-9

   1. Initial program 31.3%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

      1. add-sqr-sqrt 61.5%

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

      2. sqrt-unprod 69.5%

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

      3. pow2 69.5%

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

      4. add-sqr-sqrt 69.5%

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

      5. pow2 69.5%

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

      6. sqrt-div 69.5%

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

      7. unpow2 69.5%

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

      8. sqrt-prod 39.7%

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

      9. add-sqr-sqrt 69.5%

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

      10. unpow2 69.5%

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

      11. sqrt-prod 63.1%

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

      12. add-sqr-sqrt 69.5%

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

      13. tan-quot 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. unpow2 69.5%

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

      2. rem-sqrt-square 74.1%

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

   8. Simplified 74.1%

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

   9. Taylor expanded in eps around 0 74.1%

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

       1. rem-cbrt-cube 67.3%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.6%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.6%

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

       4. fabs-sqr 65.6%

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

       5. sqr-pow 65.6%

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

       6. unpow1/3 75.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.5%

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

       8. fma-udef 99.6%

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

   11. Simplified 99.6%

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

   if 4.5999999999999998e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{1}{t_0} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -4.6e-9)
     (+
      (/ (tan eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x)))))
      (* (tan x) (+ (/ 1.0 t_0) -1.0)))
     (if (<= eps 2.6e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ (+ (tan eps) (tan x)) t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = (tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))))) + (tan(x) * ((1.0 / t_0) + -1.0));
	} else if (eps <= 2.6e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = ((tan(eps) + tan(x)) / t_0) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -4.6e-9)
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x))))) + Float64(tan(x) * Float64(Float64(1.0 / t_0) + -1.0)));
	elseif (eps <= 2.6e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / t_0) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.6e-9], N[(N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.6e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5999999999999998e-9

   1. Initial program 47.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.1%

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

      1. tan-quot 99.3%

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

      2. expm1-log1p-u 75.3%

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

      3. expm1-udef 74.8%

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

   10. Applied egg-rr 74.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
   11. Step-by-step derivation

       1. expm1-def 75.3%

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

       2. expm1-log1p 99.3%

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

   12. Simplified 99.3%

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

       1. tan-quot 99.3%

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

       2. tan-quot 99.4%

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

       3. div-inv 99.4%

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

       4. fma-neg 99.4%

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

       5. tan-quot 99.5%

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

       6. tan-quot 99.5%

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

       7. *-commutative 99.5%

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

       8. *-commutative 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. fma-udef 99.5%

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

       2. *-rgt-identity 99.5%

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

       3. distribute-rgt-neg-in 99.5%

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

       4. metadata-eval 99.5%

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

       5. distribute-lft-out 99.5%

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

       6. *-commutative 99.5%

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

   16. Simplified 99.5%

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

   if -4.5999999999999998e-9 < eps < 2.6000000000000001e-9

   1. Initial program 31.1%

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

   2. Taylor expanded in eps around 0 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. *-lft-identity 99.1%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

      1. add-sqr-sqrt 61.0%

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

      2. sqrt-unprod 69.1%

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

      3. pow2 69.1%

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

      4. add-sqr-sqrt 69.1%

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

      5. pow2 69.1%

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

      6. sqrt-div 69.1%

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

      7. unpow2 69.1%

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

      8. sqrt-prod 39.5%

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

      9. add-sqr-sqrt 69.1%

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

      10. unpow2 69.1%

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

      11. sqrt-prod 62.6%

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

      12. add-sqr-sqrt 69.1%

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

      13. tan-quot 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. unpow2 69.1%

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

      2. rem-sqrt-square 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in eps around 0 73.6%

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

       1. rem-cbrt-cube 66.9%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.0%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.0%

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

       4. fabs-sqr 65.0%

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

       5. sqr-pow 65.0%

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

       6. unpow1/3 75.8%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.2%

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

       8. fma-udef 99.3%

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

   11. Simplified 99.3%

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

   if 2.6000000000000001e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -4.6e-9)
     (fma t_0 (/ 1.0 t_1) (- (tan x)))
     (if (<= eps 2.05e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	} else if (eps <= 2.05e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / t_1) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -4.6e-9)
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	elseif (eps <= 2.05e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.6e-9], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.05e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $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 < -4.5999999999999998e-9

   1. Initial program 47.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -4.5999999999999998e-9 < eps < 2.0500000000000002e-9

   1. Initial program 31.1%

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

   2. Taylor expanded in eps around 0 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. *-lft-identity 99.1%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

      1. add-sqr-sqrt 61.0%

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

      2. sqrt-unprod 69.1%

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

      3. pow2 69.1%

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

      4. add-sqr-sqrt 69.1%

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

      5. pow2 69.1%

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

      6. sqrt-div 69.1%

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

      7. unpow2 69.1%

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

      8. sqrt-prod 39.5%

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

      9. add-sqr-sqrt 69.1%

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

      10. unpow2 69.1%

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

      11. sqrt-prod 62.6%

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

      12. add-sqr-sqrt 69.1%

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

      13. tan-quot 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. unpow2 69.1%

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

      2. rem-sqrt-square 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in eps around 0 73.6%

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

       1. rem-cbrt-cube 66.9%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.0%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.0%

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

       4. fabs-sqr 65.0%

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

       5. sqr-pow 65.0%

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

       6. unpow1/3 75.8%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.2%

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

       8. fma-udef 99.3%

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

   11. Simplified 99.3%

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

   if 2.0500000000000002e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \tan x, \frac{1}{1 - \tan \varepsilon \cdot \tan x}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 7: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.4e-9) (not (<= eps 1.8e-9)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (tan x))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.4e-9) || !(eps <= 1.8e-9)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.4e-9) || !(eps <= 1.8e-9))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.4e-9], N[Not[LessEqual[eps, 1.8e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.3999999999999998e-9 or 1.8e-9 < eps

   1. Initial program 52.3%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.1%

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

      3. fma-neg 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. fma-neg 99.1%

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

      2. associate-*r/ 99.2%

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

      3. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   if -3.3999999999999998e-9 < eps < 1.8e-9

   1. Initial program 31.3%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

      1. add-sqr-sqrt 61.5%

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

      2. sqrt-unprod 69.5%

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

      3. pow2 69.5%

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

      4. add-sqr-sqrt 69.5%

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

      5. pow2 69.5%

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

      6. sqrt-div 69.5%

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

      7. unpow2 69.5%

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

      8. sqrt-prod 39.7%

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

      9. add-sqr-sqrt 69.5%

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

      10. unpow2 69.5%

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

      11. sqrt-prod 63.1%

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

      12. add-sqr-sqrt 69.5%

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

      13. tan-quot 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. unpow2 69.5%

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

      2. rem-sqrt-square 74.1%

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

   8. Simplified 74.1%

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

   9. Taylor expanded in eps around 0 74.1%

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

       1. rem-cbrt-cube 67.3%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.6%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.6%

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

       4. fabs-sqr 65.6%

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

       5. sqr-pow 65.6%

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

       6. unpow1/3 75.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.5%

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

       8. fma-udef 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 8: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1 - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))) (t_1 (+ (tan eps) (tan x))))
   (if (<= eps -5.2e-9)
     (- (* (/ 1.0 t_0) t_1) (tan x))
     (if (<= eps 2.4e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_1 t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double t_1 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -5.2e-9) {
		tmp = ((1.0 / t_0) * t_1) - tan(x);
	} else if (eps <= 2.4e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_1 / t_0) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_1 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -5.2e-9)
		tmp = Float64(Float64(Float64(1.0 / t_0) * t_1) - tan(x));
	elseif (eps <= 2.4e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_1 / t_0) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.2e-9], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.4e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.2000000000000002e-9

   1. Initial program 47.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -5.2000000000000002e-9 < eps < 2.4e-9

   1. Initial program 31.1%

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

   2. Taylor expanded in eps around 0 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. *-lft-identity 99.1%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

      1. add-sqr-sqrt 61.0%

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

      2. sqrt-unprod 69.1%

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

      3. pow2 69.1%

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

      4. add-sqr-sqrt 69.1%

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

      5. pow2 69.1%

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

      6. sqrt-div 69.1%

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

      7. unpow2 69.1%

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

      8. sqrt-prod 39.5%

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

      9. add-sqr-sqrt 69.1%

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

      10. unpow2 69.1%

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

      11. sqrt-prod 62.6%

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

      12. add-sqr-sqrt 69.1%

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

      13. tan-quot 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. unpow2 69.1%

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

      2. rem-sqrt-square 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in eps around 0 73.6%

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

       1. rem-cbrt-cube 66.9%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.0%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.0%

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

       4. fabs-sqr 65.0%

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

       5. sqr-pow 65.0%

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

       6. unpow1/3 75.8%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.2%

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

       8. fma-udef 99.3%

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

   11. Simplified 99.3%

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

   if 2.4e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 - \tan \varepsilon \cdot \tan x} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 9: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -2.1e-9)
     (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 3e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 (- 1.0 (* (tan eps) (tan x)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -2.1e-9) {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 3e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -2.1e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 3e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.1e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-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[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.10000000000000019e-9

   1. Initial program 47.0%

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

      1. tan-sum 98.8%

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

      2. div-inv 98.8%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. fma-neg 98.8%

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

      2. associate-*r/ 98.8%

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

      3. *-rgt-identity 98.8%

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

   5. Simplified 98.8%

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

      1. tan-quot 98.7%

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

      2. clear-num 98.7%

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

      3. un-div-inv 98.7%

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

      4. clear-num 98.7%

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

      5. tan-quot 98.8%

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

   7. Applied egg-rr 98.8%

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

   if -2.10000000000000019e-9 < eps < 2.99999999999999998e-9

   1. Initial program 31.3%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

      1. add-sqr-sqrt 61.5%

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

      2. sqrt-unprod 69.5%

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

      3. pow2 69.5%

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

      4. add-sqr-sqrt 69.5%

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

      5. pow2 69.5%

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

      6. sqrt-div 69.5%

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

      7. unpow2 69.5%

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

      8. sqrt-prod 39.7%

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

      9. add-sqr-sqrt 69.5%

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

      10. unpow2 69.5%

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

      11. sqrt-prod 63.1%

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

      12. add-sqr-sqrt 69.5%

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

      13. tan-quot 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. unpow2 69.5%

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

      2. rem-sqrt-square 74.1%

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

   8. Simplified 74.1%

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

   9. Taylor expanded in eps around 0 74.1%

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

       1. rem-cbrt-cube 67.3%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.6%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.6%

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

       4. fabs-sqr 65.6%

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

       5. sqr-pow 65.6%

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

       6. unpow1/3 75.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.5%

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

       8. fma-udef 99.6%

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

   11. Simplified 99.6%

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

   if 2.99999999999999998e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 10: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.8e-8)
   (tan eps)
   (if (<= eps 3.75e-6) (fma eps (pow (tan x) 2.0) eps) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = tan(eps);
	} else if (eps <= 3.75e-6) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5.8e-8)
		tmp = tan(eps);
	elseif (eps <= 3.75e-6)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = tan(eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5.8e-8], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.75e-6], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.8000000000000003e-8 or 3.7500000000000001e-6 < eps

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 53.4%

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

      1. tan-quot 99.4%

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

      2. expm1-log1p-u 72.3%

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

      3. expm1-udef 71.9%

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

   4. Applied egg-rr 38.7%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 99.4%

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

   6. Simplified 53.5%

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

   if -5.8000000000000003e-8 < eps < 3.7500000000000001e-6

   1. Initial program 31.1%

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

   2. Taylor expanded in eps around 0 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. *-lft-identity 99.1%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

      1. add-sqr-sqrt 61.0%

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

      2. sqrt-unprod 69.1%

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

      3. pow2 69.1%

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

      4. add-sqr-sqrt 69.1%

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

      5. pow2 69.1%

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

      6. sqrt-div 69.1%

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

      7. unpow2 69.1%

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

      8. sqrt-prod 39.5%

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

      9. add-sqr-sqrt 69.1%

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

      10. unpow2 69.1%

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

      11. sqrt-prod 62.6%

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

      12. add-sqr-sqrt 69.1%

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

      13. tan-quot 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. unpow2 69.1%

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

      2. rem-sqrt-square 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in eps around 0 73.6%

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

       1. rem-cbrt-cube 66.9%

          \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}}\right| + \varepsilon \]

       2. unpow1/3 65.0%

          \[\leadsto \left|\color{blue}{{\left({\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right| + \varepsilon \]

       3. sqr-pow 65.0%

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

       4. fabs-sqr 65.0%

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

       5. sqr-pow 65.0%

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

       6. unpow1/3 75.8%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\varepsilon \cdot {\tan x}^{2}\right)}^{3}}} + \varepsilon \]

       7. rem-cbrt-cube 99.2%

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

       8. fma-udef 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.8e-8)
   (tan eps)
   (if (<= eps 3.7e-6) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = tan(eps);
	} else if (eps <= 3.7e-6) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-5.8d-8)) then
        tmp = tan(eps)
    else if (eps <= 3.7d-6) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -5.8e-8) {
		tmp = Math.tan(eps);
	} else if (eps <= 3.7e-6) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -5.8e-8:
		tmp = math.tan(eps)
	elif eps <= 3.7e-6:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5.8e-8)
		tmp = tan(eps);
	elseif (eps <= 3.7e-6)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -5.8e-8)
		tmp = tan(eps);
	elseif (eps <= 3.7e-6)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5.8e-8], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.7e-6], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.8000000000000003e-8 or 3.7000000000000002e-6 < eps

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 53.4%

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

      1. tan-quot 99.4%

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

      2. expm1-log1p-u 72.3%

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

      3. expm1-udef 71.9%

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

   4. Applied egg-rr 38.7%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 99.4%

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

   6. Simplified 53.5%

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

   if -5.8000000000000003e-8 < eps < 3.7000000000000002e-6

   1. Initial program 31.1%

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

   2. Taylor expanded in eps around 0 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. *-lft-identity 99.1%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 63.6%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

   6. Applied egg-rr 63.6%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.2%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.6%

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

2. Taylor expanded in x around 0 57.8%

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

   1. tan-quot 82.8%

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

   2. expm1-log1p-u 68.2%

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

   3. expm1-udef 42.0%

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

4. Applied egg-rr 23.9%

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

   1. expm1-def 68.2%

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

   2. expm1-log1p 82.8%

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

6. Simplified 57.9%

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

7. Final simplification 57.9%

   \[\leadsto \tan \varepsilon \]

Alternative 13: 4.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 42.6%

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

   1. add-cube-cbrt 41.7%

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

   2. pow3 41.7%

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

   3. add-exp-log 18.4%

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

   4. pow-exp 18.3%

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

3. Applied egg-rr 18.3%

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

4. Taylor expanded in eps around 0 4.2%

   \[\leadsto \color{blue}{\frac{\sin x}{\cos x} \cdot {1}^{0.3333333333333333} - \frac{\sin x}{\cos x}} \]
5. Step-by-step derivation

   1. pow-base-1 4.2%

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

   2. *-rgt-identity 4.2%

      \[\leadsto \color{blue}{\frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x} \]

   3. +-inverses 4.2%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.2%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.2%

   \[\leadsto 0 \]

Alternative 14: 30.7% 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 42.6%

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

2. Taylor expanded in x around 0 57.8%

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

3. Taylor expanded in eps around 0 31.3%

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

4. Final simplification 31.3%

   \[\leadsto \varepsilon \]

Developer target: 76.3% 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 2023240 
(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)))