2tan (problem 3.3.2)

Percentage Accurate: 42.3% → 99.5%

Time: 19.4s

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: 42.3% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{-\sin \varepsilon}{-\cos \varepsilon}, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.5e-7)
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (/ -1.0 (/ -1.0 (tan eps))))))
    (tan x))
   (if (<= eps 1.55e-7)
     (+
      (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
      (*
       (* eps eps)
       (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
     (-
      (/
       (- (- (tan x)) (tan eps))
       (fma (tan x) (/ (- (sin eps)) (- (cos eps))) -1.0))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.5e-7) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	} else if (eps <= 1.55e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = ((-tan(x) - tan(eps)) / fma(tan(x), (-sin(eps) / -cos(eps)), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.5e-7)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * Float64(-1.0 / Float64(-1.0 / tan(eps)))))) - tan(x));
	elseif (eps <= 1.55e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), Float64(Float64(-sin(eps)) / Float64(-cos(eps))), -1.0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.5e-7], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.55e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[((-N[Sin[eps], $MachinePrecision]) / (-N[Cos[eps], $MachinePrecision])), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.4999999999999999e-7

   1. Initial program 54.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      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. Step-by-step derivation

      1. tan-quot 99.5%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.5%

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

      4. clear-num 99.5%

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

      5. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. distribute-neg-frac 99.5%

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

       2. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

   if -1.4999999999999999e-7 < eps < 1.55e-7

   1. Initial program 29.8%

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

      1. tan-sum 31.2%

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

      2. div-inv 31.2%

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

      3. fma-neg 31.2%

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

   3. Applied egg-rr 31.2%

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

      1. fma-neg 31.2%

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

      2. associate-*r/ 31.2%

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

      3. *-rgt-identity 31.2%

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

   5. Simplified 31.2%

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

      1. *-commutative 31.2%

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

      2. tan-quot 31.2%

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

      3. clear-num 31.2%

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

      4. tan-quot 31.2%

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

      5. frac-times 31.2%

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

      6. *-un-lft-identity 31.2%

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

      7. clear-num 31.2%

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

      8. tan-quot 31.2%

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

   7. Applied egg-rr 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-*r/ 31.2%

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

      3. *-rgt-identity 31.2%

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

   9. Simplified 31.2%

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

   10. Taylor expanded in eps around 0 99.6%

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

       1. mul-1-neg 99.6%

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

       2. unsub-neg 99.6%

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

       3. cancel-sign-sub-inv 99.6%

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

       4. metadata-eval 99.6%

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

       5. *-lft-identity 99.6%

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

       6. distribute-lft-in 99.7%

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

       7. *-rgt-identity 99.7%

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

       8. unpow2 99.7%

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

   12. Simplified 99.7%

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

   if 1.55e-7 < eps

   1. Initial program 58.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.5%

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

      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. Step-by-step derivation

      1. frac-2neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. add-sqr-sqrt 56.1%

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

      8. sqrt-unprod 84.5%

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

      9. sqr-neg 84.5%

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

      10. sqrt-unprod 28.4%

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

      11. add-sqr-sqrt 61.3%

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

      12. distribute-lft-neg-in 61.3%

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

      13. add-sqr-sqrt 32.9%

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

      14. sqrt-unprod 76.3%

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

      15. sqr-neg 76.3%

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

      16. sqrt-unprod 43.3%

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

      17. add-sqr-sqrt 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. distribute-neg-frac 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

       1. tan-quot 99.6%

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

       2. frac-2neg 99.6%

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{-\sin \varepsilon}{-\cos \varepsilon}, -1\right)} - \tan x\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{-\sin \varepsilon}{-\cos \varepsilon}, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.65e-7)
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (/ -1.0 (/ -1.0 (tan eps))))))
    (tan x))
   (if (<= eps 4.5e-8)
     (+
      (+ eps (* eps (* (pow (sin x) 2.0) (pow (cos x) -2.0))))
      (*
       (* eps eps)
       (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
     (-
      (/
       (- (- (tan x)) (tan eps))
       (fma (tan x) (/ (- (sin eps)) (- (cos eps))) -1.0))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.65e-7) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	} else if (eps <= 4.5e-8) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) * pow(cos(x), -2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = ((-tan(x) - tan(eps)) / fma(tan(x), (-sin(eps) / -cos(eps)), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2.65e-7)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * Float64(-1.0 / Float64(-1.0 / tan(eps)))))) - tan(x));
	elseif (eps <= 4.5e-8)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) * (cos(x) ^ -2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), Float64(Float64(-sin(eps)) / Float64(-cos(eps))), -1.0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.65e-7], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-8], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[((-N[Sin[eps], $MachinePrecision]) / (-N[Cos[eps], $MachinePrecision])), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.65e-7

   1. Initial program 54.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      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. Step-by-step derivation

      1. tan-quot 99.5%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.5%

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

      4. clear-num 99.5%

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

      5. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. distribute-neg-frac 99.5%

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

       2. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

   if -2.65e-7 < eps < 4.49999999999999993e-8

   1. Initial program 29.8%

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

      1. tan-sum 31.2%

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

      2. div-inv 31.2%

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

      3. fma-neg 31.2%

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

   3. Applied egg-rr 31.2%

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

      1. fma-neg 31.2%

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

      2. associate-*r/ 31.2%

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

      3. *-rgt-identity 31.2%

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

   5. Simplified 31.2%

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

   6. Taylor expanded in eps around 0 99.6%

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

   8. Simplified 99.7%

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

   if 4.49999999999999993e-8 < eps

   1. Initial program 58.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.5%

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

      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. Step-by-step derivation

      1. frac-2neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. add-sqr-sqrt 56.1%

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

      8. sqrt-unprod 84.5%

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

      9. sqr-neg 84.5%

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

      10. sqrt-unprod 28.4%

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

      11. add-sqr-sqrt 61.3%

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

      12. distribute-lft-neg-in 61.3%

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

      13. add-sqr-sqrt 32.9%

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

      14. sqrt-unprod 76.3%

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

      15. sqr-neg 76.3%

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

      16. sqrt-unprod 43.3%

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

      17. add-sqr-sqrt 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. distribute-neg-frac 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

       1. tan-quot 99.6%

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

       2. frac-2neg 99.6%

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{-\sin \varepsilon}{-\cos \varepsilon}, -1\right)} - \tan x\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{-\sin \varepsilon}{-\cos \varepsilon}, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5e-9)
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (/ -1.0 (/ -1.0 (tan eps))))))
    (tan x))
   (if (<= eps 4.4e-9)
     (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))
     (-
      (/
       (- (- (tan x)) (tan eps))
       (fma (tan x) (/ (- (sin eps)) (- (cos eps))) -1.0))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5e-9) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	} else if (eps <= 4.4e-9) {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	} else {
		tmp = ((-tan(x) - tan(eps)) / fma(tan(x), (-sin(eps) / -cos(eps)), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5e-9)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * Float64(-1.0 / Float64(-1.0 / tan(eps)))))) - tan(x));
	elseif (eps <= 4.4e-9)
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), Float64(Float64(-sin(eps)) / Float64(-cos(eps))), -1.0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5e-9], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.4e-9], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[((-N[Sin[eps], $MachinePrecision]) / (-N[Cos[eps], $MachinePrecision])), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.0000000000000001e-9

   1. Initial program 53.2%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.0%

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

      3. 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 99.0%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

      1. tan-quot 99.0%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.0%

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

      4. clear-num 99.0%

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

      5. tan-quot 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. frac-2neg 99.0%

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

      2. div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. distribute-neg-frac 99.0%

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

       2. metadata-eval 99.0%

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

   11. Simplified 99.0%

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

   if -5.0000000000000001e-9 < eps < 4.3999999999999997e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   if 4.3999999999999997e-9 < eps

   1. Initial program 58.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.5%

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

      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. Step-by-step derivation

      1. frac-2neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. add-sqr-sqrt 56.1%

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

      8. sqrt-unprod 84.5%

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

      9. sqr-neg 84.5%

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

      10. sqrt-unprod 28.4%

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

      11. add-sqr-sqrt 61.3%

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

      12. distribute-lft-neg-in 61.3%

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

      13. add-sqr-sqrt 32.9%

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

      14. sqrt-unprod 76.3%

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

      15. sqr-neg 76.3%

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

      16. sqrt-unprod 43.3%

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

      17. add-sqr-sqrt 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. distribute-neg-frac 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

       1. tan-quot 99.6%

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

       2. frac-2neg 99.6%

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{-\sin \varepsilon}{-\cos \varepsilon}, -1\right)} - \tan x\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \frac{1}{\frac{1}{\tan \varepsilon}}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.7e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (/ -1.0 (/ -1.0 (tan eps)))))) (tan x))
     (if (<= eps 3.5e-9)
       (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))
       (- (- (tan x)) (/ t_0 (fma (tan x) (/ 1.0 (/ 1.0 (tan eps))) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.7e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	} else if (eps <= 3.5e-9) {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), (1.0 / (1.0 / tan(eps))), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.7e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * Float64(-1.0 / Float64(-1.0 / tan(eps)))))) - tan(x));
	elseif (eps <= 3.5e-9)
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), Float64(1.0 / Float64(1.0 / tan(eps))), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.7e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.5e-9], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[(1.0 / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.6999999999999999e-9

   1. Initial program 53.2%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.0%

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

      3. 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 99.0%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

      1. tan-quot 99.0%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.0%

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

      4. clear-num 99.0%

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

      5. tan-quot 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. frac-2neg 99.0%

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

      2. div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. distribute-neg-frac 99.0%

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

       2. metadata-eval 99.0%

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

   11. Simplified 99.0%

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

   if -4.6999999999999999e-9 < eps < 3.4999999999999999e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   if 3.4999999999999999e-9 < eps

   1. Initial program 58.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.5%

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

      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. Step-by-step derivation

      1. frac-2neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. add-sqr-sqrt 56.1%

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

      8. sqrt-unprod 84.5%

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

      9. sqr-neg 84.5%

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

      10. sqrt-unprod 28.4%

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

      11. add-sqr-sqrt 61.3%

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

      12. distribute-lft-neg-in 61.3%

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

      13. add-sqr-sqrt 32.9%

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

      14. sqrt-unprod 76.3%

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

      15. sqr-neg 76.3%

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

      16. sqrt-unprod 43.3%

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

      17. add-sqr-sqrt 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. distribute-neg-frac 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

       1. remove-double-div 99.5%

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

   11. Applied egg-rr 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \frac{1}{\frac{1}{\tan \varepsilon}}, -1\right)}\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \sin x \cdot \frac{\tan \varepsilon}{\cos x}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.7e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (/ -1.0 (/ -1.0 (tan eps)))))) (tan x))
     (if (<= eps 4.4e-9)
       (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))
       (- (/ t_0 (- 1.0 (* (sin x) (/ (tan eps) (cos x))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	} else if (eps <= 4.4e-9) {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - (sin(x) * (tan(eps) / cos(x))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-3.7d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) * ((-1.0d0) / ((-1.0d0) / tan(eps)))))) - tan(x)
    else if (eps <= 4.4d-9) then
        tmp = eps + (eps * ((0.5d0 - (cos((x + x)) / 2.0d0)) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 / (1.0d0 - (sin(x) * (tan(eps) / cos(x))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) * (-1.0 / (-1.0 / Math.tan(eps)))))) - Math.tan(x);
	} else if (eps <= 4.4e-9) {
		tmp = eps + (eps * ((0.5 - (Math.cos((x + x)) / 2.0)) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - (Math.sin(x) * (Math.tan(eps) / Math.cos(x))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -3.7e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) * (-1.0 / (-1.0 / math.tan(eps)))))) - math.tan(x)
	elif eps <= 4.4e-9:
		tmp = eps + (eps * ((0.5 - (math.cos((x + x)) / 2.0)) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 / (1.0 - (math.sin(x) * (math.tan(eps) / math.cos(x))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.7e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * Float64(-1.0 / Float64(-1.0 / tan(eps)))))) - tan(x));
	elseif (eps <= 4.4e-9)
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(sin(x) * Float64(tan(eps) / cos(x))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -3.7e-9)
		tmp = (t_0 / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	elseif (eps <= 4.4e-9)
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 / (1.0 - (sin(x) * (tan(eps) / cos(x))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.7e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.4e-9], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.7e-9

   1. Initial program 53.2%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.0%

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

      3. 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 99.0%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

      1. tan-quot 99.0%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.0%

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

      4. clear-num 99.0%

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

      5. tan-quot 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. frac-2neg 99.0%

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

      2. div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. distribute-neg-frac 99.0%

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

       2. metadata-eval 99.0%

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

   11. Simplified 99.0%

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

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

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   if 4.3999999999999997e-9 < eps

   1. Initial program 58.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.5%

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

      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. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.4%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.5%

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

      8. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   9. Simplified 99.5%

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

       1. div-inv 99.5%

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

       2. clear-num 99.5%

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

   11. Applied egg-rr 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sin x \cdot \frac{\tan \varepsilon}{\cos x}} - \tan x\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.2e-9) (not (<= eps 4.4e-9)))
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (/ -1.0 (/ -1.0 (tan eps))))))
    (tan x))
   (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.2e-9) || !(eps <= 4.4e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	} else {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.2d-9)) .or. (.not. (eps <= 4.4d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * ((-1.0d0) / ((-1.0d0) / tan(eps)))))) - tan(x)
    else
        tmp = eps + (eps * ((0.5d0 - (cos((x + x)) / 2.0d0)) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.2e-9) || !(eps <= 4.4e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * (-1.0 / (-1.0 / Math.tan(eps)))))) - Math.tan(x);
	} else {
		tmp = eps + (eps * ((0.5 - (Math.cos((x + x)) / 2.0)) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.2e-9) or not (eps <= 4.4e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * (-1.0 / (-1.0 / math.tan(eps)))))) - math.tan(x)
	else:
		tmp = eps + (eps * ((0.5 - (math.cos((x + x)) / 2.0)) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.2e-9) || !(eps <= 4.4e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * Float64(-1.0 / Float64(-1.0 / tan(eps)))))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.2e-9) || ~((eps <= 4.4e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * (-1.0 / (-1.0 / tan(eps)))))) - tan(x);
	else
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.2e-9], N[Not[LessEqual[eps, 4.4e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[(-1.0 / N[(-1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.20000000000000039e-9 or 4.3999999999999997e-9 < eps

   1. Initial program 55.9%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.1%

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

      3. fma-neg 99.1%

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

   3. Applied egg-rr 99.1%

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

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

      1. tan-quot 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.2%

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

      4. clear-num 99.2%

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

      5. tan-quot 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. frac-2neg 99.2%

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

      2. div-inv 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. distribute-neg-frac 99.3%

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

       2. metadata-eval 99.3%

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

   11. Simplified 99.3%

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

   if -4.20000000000000039e-9 < eps < 4.3999999999999997e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \frac{-1}{\frac{-1}{\tan \varepsilon}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -5e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 5.2e-9)
       (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -5e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 5.2e-9) {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-5d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 5.2d-9) then
        tmp = eps + (eps * ((0.5d0 - (cos((x + x)) / 2.0d0)) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -5e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 5.2e-9) {
		tmp = eps + (eps * ((0.5 - (Math.cos((x + x)) / 2.0)) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -5e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 5.2e-9:
		tmp = eps + (eps * ((0.5 - (math.cos((x + x)) / 2.0)) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -5e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 5.2e-9)
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -5e-9)
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 5.2e-9)
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-9], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.0000000000000001e-9

   1. Initial program 53.2%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.0%

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

      3. 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 99.0%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

   if -5.0000000000000001e-9 < eps < 5.2000000000000002e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   if 5.2000000000000002e-9 < eps

   1. Initial program 58.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.3%

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

      2. associate-*r/ 99.5%

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

      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. Step-by-step derivation

      1. tan-quot 99.5%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.5%

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

      4. clear-num 99.5%

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

      5. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 8: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.7e-9) (not (<= eps 3.6e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-9) || !(eps <= 3.6e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.7d-9)) .or. (.not. (eps <= 3.6d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((0.5d0 - (cos((x + x)) / 2.0d0)) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-9) || !(eps <= 3.6e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * ((0.5 - (Math.cos((x + x)) / 2.0)) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.7e-9) or not (eps <= 3.6e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * ((0.5 - (math.cos((x + x)) / 2.0)) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.7e-9) || !(eps <= 3.6e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.7e-9) || ~((eps <= 3.6e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.7e-9], N[Not[LessEqual[eps, 3.6e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.7e-9 or 3.6e-9 < eps

   1. Initial program 55.9%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.1%

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

      3. fma-neg 99.1%

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

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
   4. 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 -3.7e-9 < eps < 3.6e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 9: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.5e-9)
   (-
    (/
     (+ (tan x) (tan eps))
     (- 1.0 (/ (tan x) (+ (/ 1.0 eps) (* eps -0.3333333333333333)))))
    (tan x))
   (if (<= eps 5.2e-7)
     (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5.5e-9) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / ((1.0 / eps) + (eps * -0.3333333333333333))))) - tan(x);
	} else if (eps <= 5.2e-7) {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-5.5d-9)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) / ((1.0d0 / eps) + (eps * (-0.3333333333333333d0)))))) - tan(x)
    else if (eps <= 5.2d-7) then
        tmp = eps + (eps * ((0.5d0 - (cos((x + x)) / 2.0d0)) / (cos(x) ** 2.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -5.5e-9) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) / ((1.0 / eps) + (eps * -0.3333333333333333))))) - Math.tan(x);
	} else if (eps <= 5.2e-7) {
		tmp = eps + (eps * ((0.5 - (Math.cos((x + x)) / 2.0)) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -5.5e-9:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) / ((1.0 / eps) + (eps * -0.3333333333333333))))) - math.tan(x)
	elif eps <= 5.2e-7:
		tmp = eps + (eps * ((0.5 - (math.cos((x + x)) / 2.0)) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5.5e-9)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) / Float64(Float64(1.0 / eps) + Float64(eps * -0.3333333333333333))))) - tan(x));
	elseif (eps <= 5.2e-7)
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -5.5e-9)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / ((1.0 / eps) + (eps * -0.3333333333333333))))) - tan(x);
	elseif (eps <= 5.2e-7)
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / (cos(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5.5e-9], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[(1.0 / eps), $MachinePrecision] + N[(eps * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-7], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.4999999999999996e-9

   1. Initial program 53.2%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.0%

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

      3. 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 99.0%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

   5. Simplified 99.0%

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

      1. tan-quot 99.0%

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

      2. clear-num 99.1%

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

      3. un-div-inv 99.0%

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

      4. clear-num 99.0%

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

      5. tan-quot 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in eps around 0 58.8%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\color{blue}{\frac{1}{\varepsilon} + -0.3333333333333333 \cdot \varepsilon}}} - \tan x \]
   9. Step-by-step derivation

      1. *-commutative 58.8%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \color{blue}{\varepsilon \cdot -0.3333333333333333}}} - \tan x \]

   10. Simplified 58.8%

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\color{blue}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}}} - \tan x \]

   if -5.4999999999999996e-9 < eps < 5.19999999999999998e-7

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. +-inverses 99.3%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 99.3%

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

      4. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   if 5.19999999999999998e-7 < eps

   1. Initial program 58.9%

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

   2. Taylor expanded in x around 0 61.4%

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

      1. tan-quot 61.7%

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

      2. expm1-log1p-u 46.5%

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

      3. expm1-udef 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. expm1-def 46.5%

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

      2. expm1-log1p 61.7%

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

   6. Simplified 61.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 10: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.062:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.062)
   (tan eps)
   (if (<= eps 5.2e-7)
     (+ eps (* eps (/ (- 0.5 (/ (cos (+ x x)) 2.0)) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.062) {
		tmp = tan(eps);
	} else if (eps <= 5.2e-7) {
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.062d0)) then
        tmp = tan(eps)
    else if (eps <= 5.2d-7) then
        tmp = eps + (eps * ((0.5d0 - (cos((x + x)) / 2.0d0)) / (cos(x) ** 2.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.062) {
		tmp = Math.tan(eps);
	} else if (eps <= 5.2e-7) {
		tmp = eps + (eps * ((0.5 - (Math.cos((x + x)) / 2.0)) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.062:
		tmp = math.tan(eps)
	elif eps <= 5.2e-7:
		tmp = eps + (eps * ((0.5 - (math.cos((x + x)) / 2.0)) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.062)
		tmp = tan(eps);
	elseif (eps <= 5.2e-7)
		tmp = Float64(eps + Float64(eps * Float64(Float64(0.5 - Float64(cos(Float64(x + x)) / 2.0)) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.062)
		tmp = tan(eps);
	elseif (eps <= 5.2e-7)
		tmp = eps + (eps * ((0.5 - (cos((x + x)) / 2.0)) / (cos(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.062], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 5.2e-7], N[(eps + N[(eps * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.062 or 5.19999999999999998e-7 < eps

   1. Initial program 56.7%

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

   2. Taylor expanded in x around 0 59.9%

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

      1. tan-quot 60.2%

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

      2. expm1-log1p-u 43.8%

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

      3. expm1-udef 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. expm1-def 43.8%

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

      2. expm1-log1p 60.2%

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

   6. Simplified 60.2%

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

   if -0.062 < eps < 5.19999999999999998e-7

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.3%

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

      1. cancel-sign-sub-inv 98.3%

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

      2. metadata-eval 98.3%

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

      3. *-lft-identity 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. *-rgt-identity 98.3%

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

   4. Simplified 98.3%

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

      1. unpow2 98.3%

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

      2. sin-mult 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

         \[\leadsto \varepsilon + \varepsilon \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}} \]

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

   8. Simplified 98.4%

      \[\leadsto \varepsilon + \varepsilon \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.062:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{0.5 - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.062:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon}{\frac{1}{{\tan x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.062)
   (tan eps)
   (if (<= eps 5.2e-7) (+ eps (/ eps (/ 1.0 (pow (tan x) 2.0)))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.062) {
		tmp = tan(eps);
	} else if (eps <= 5.2e-7) {
		tmp = eps + (eps / (1.0 / 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 <= (-0.062d0)) then
        tmp = tan(eps)
    else if (eps <= 5.2d-7) then
        tmp = eps + (eps / (1.0d0 / (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 <= -0.062) {
		tmp = Math.tan(eps);
	} else if (eps <= 5.2e-7) {
		tmp = eps + (eps / (1.0 / Math.pow(Math.tan(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.062:
		tmp = math.tan(eps)
	elif eps <= 5.2e-7:
		tmp = eps + (eps / (1.0 / math.pow(math.tan(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.062)
		tmp = tan(eps);
	elseif (eps <= 5.2e-7)
		tmp = Float64(eps + Float64(eps / Float64(1.0 / (tan(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.062)
		tmp = tan(eps);
	elseif (eps <= 5.2e-7)
		tmp = eps + (eps / (1.0 / (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.062], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 5.2e-7], N[(eps + N[(eps / N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.062 or 5.19999999999999998e-7 < eps

   1. Initial program 56.7%

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

   2. Taylor expanded in x around 0 59.9%

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

      1. tan-quot 60.2%

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

      2. expm1-log1p-u 43.8%

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

      3. expm1-udef 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. expm1-def 43.8%

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

      2. expm1-log1p 60.2%

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

   6. Simplified 60.2%

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

   if -0.062 < eps < 5.19999999999999998e-7

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.3%

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

      1. cancel-sign-sub-inv 98.3%

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

      2. metadata-eval 98.3%

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

      3. *-lft-identity 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. *-rgt-identity 98.3%

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

   4. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. un-div-inv 98.3%

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

      3. clear-num 98.3%

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

      4. unpow2 98.3%

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

      5. unpow2 98.3%

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

      6. frac-times 98.3%

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

      7. tan-quot 98.3%

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

      8. tan-quot 98.3%

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

      9. pow2 98.3%

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

   6. Applied egg-rr 98.3%

      \[\leadsto \varepsilon + \color{blue}{\frac{\varepsilon}{\frac{1}{{\tan x}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.062:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon}{\frac{1}{{\tan x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.062:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.062)
   (tan eps)
   (if (<= eps 4.6e-7) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.062) {
		tmp = tan(eps);
	} else if (eps <= 4.6e-7) {
		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 <= (-0.062d0)) then
        tmp = tan(eps)
    else if (eps <= 4.6d-7) 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 <= -0.062) {
		tmp = Math.tan(eps);
	} else if (eps <= 4.6e-7) {
		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 <= -0.062:
		tmp = math.tan(eps)
	elif eps <= 4.6e-7:
		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 <= -0.062)
		tmp = tan(eps);
	elseif (eps <= 4.6e-7)
		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 <= -0.062)
		tmp = tan(eps);
	elseif (eps <= 4.6e-7)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.062], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 4.6e-7], 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 < -0.062 or 4.5999999999999999e-7 < eps

   1. Initial program 56.7%

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

   2. Taylor expanded in x around 0 59.9%

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

      1. tan-quot 60.2%

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

      2. expm1-log1p-u 43.8%

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

      3. expm1-udef 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. expm1-def 43.8%

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

      2. expm1-log1p 60.2%

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

   6. Simplified 60.2%

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

   if -0.062 < eps < 4.5999999999999999e-7

   1. Initial program 29.6%

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

   2. Taylor expanded in eps around 0 98.3%

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

      1. cancel-sign-sub-inv 98.3%

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

      2. metadata-eval 98.3%

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

      3. *-lft-identity 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. *-rgt-identity 98.3%

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

   4. Simplified 98.3%

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

      1. expm1-log1p-u 98.3%

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

      2. expm1-udef 61.6%

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

      3. unpow2 61.6%

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

      4. unpow2 61.6%

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

      5. frac-times 61.6%

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

      6. tan-quot 61.6%

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

      7. tan-quot 61.6%

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

      8. pow2 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. expm1-def 98.3%

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

      2. expm1-log1p 98.3%

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

   8. Simplified 98.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.062:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 13: 58.3% 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 43.4%

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

2. Taylor expanded in x around 0 60.3%

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

   1. tan-quot 60.5%

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

   2. expm1-log1p-u 52.1%

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

   3. expm1-udef 24.9%

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

4. Applied egg-rr 24.9%

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

   1. expm1-def 52.1%

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

   2. expm1-log1p 60.5%

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

6. Simplified 60.5%

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

7. Final simplification 60.5%

   \[\leadsto \tan \varepsilon \]

Alternative 14: 30.8% 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 43.4%

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

2. Taylor expanded in x around 0 60.3%

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

3. Taylor expanded in eps around 0 32.0%

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

4. Final simplification 32.0%

   \[\leadsto \varepsilon \]

Developer target: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023257 
(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)))