2tan (problem 3.3.2)

Percentage Accurate: 42.3% → 99.5%

Time: 18.3s

Alternatives: 13

Speedup: 1.0×

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t_0 \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{1 - t_1 \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{\varepsilon \cdot \sin x}{\cos x} + \frac{{\varepsilon}^{2}}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (/ (sin eps) (cos eps))))
   (if (<= eps -3.8e-6)
     (fma
      (* t_0 (fma (tan x) (tan eps) 1.0))
      (/ 1.0 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
      (- (tan x)))
     (if (<= eps 2.2e-6)
       (+
        (/ t_1 (- 1.0 (* t_1 (/ (sin x) (cos x)))))
        (*
         (tan x)
         (+
          (/ (* eps (sin x)) (cos x))
          (/ (pow eps 2.0) (/ (pow (cos x) 2.0) (pow (sin x) 2.0))))))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ (cos eps) (sin eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = sin(eps) / cos(eps);
	double tmp;
	if (eps <= -3.8e-6) {
		tmp = fma((t_0 * fma(tan(x), tan(eps), 1.0)), (1.0 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), -tan(x));
	} else if (eps <= 2.2e-6) {
		tmp = (t_1 / (1.0 - (t_1 * (sin(x) / cos(x))))) + (tan(x) * (((eps * sin(x)) / cos(x)) + (pow(eps, 2.0) / (pow(cos(x), 2.0) / pow(sin(x), 2.0)))));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (cos(eps) / sin(eps))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if (eps <= -3.8e-6)
		tmp = fma(Float64(t_0 * fma(tan(x), tan(eps), 1.0)), Float64(1.0 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), Float64(-tan(x)));
	elseif (eps <= 2.2e-6)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(t_1 * Float64(sin(x) / cos(x))))) + Float64(tan(x) * Float64(Float64(Float64(eps * sin(x)) / cos(x)) + Float64((eps ^ 2.0) / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(cos(eps) / sin(eps))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.8e-6], N[(N[(t$95$0 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.2e-6], N[(N[(t$95$1 / N[(1.0 - N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.8e-6

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. associate-/r/ 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow2 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   10. Simplified 99.4%

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

       1. div-inv 99.3%

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

       2. fma-neg 99.5%

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

   12. Applied egg-rr 99.5%

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

   if -3.8e-6 < eps < 2.2000000000000001e-6

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 31.2%

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

      4. prod-diff 31.1%

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

      5. *-commutative 31.1%

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

      6. *-un-lft-identity 31.1%

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

      7. *-commutative 31.1%

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

      8. *-un-lft-identity 31.1%

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

   4. Applied egg-rr 31.1%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in x around inf 31.1%

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

      1. associate--l+ 58.2%

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

      2. associate-/r* 58.2%

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

      3. times-frac 58.2%

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

   9. Simplified 58.2%

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

       1. tan-quot 55.5%

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

       2. tan-quot 58.2%

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

       3. div-inv 58.2%

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

       4. fma-neg 58.2%

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

       5. tan-quot 58.2%

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

       6. tan-quot 58.2%

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

       7. *-commutative 58.2%

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

   11. Applied egg-rr 58.2%

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

       1. fma-udef 58.2%

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

       2. *-rgt-identity 58.2%

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

       3. distribute-rgt-neg-in 58.2%

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

       4. metadata-eval 58.2%

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

       5. distribute-lft-out 58.2%

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

   13. Simplified 58.2%

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

   14. Taylor expanded in eps around 0 99.8%

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

       1. associate-/l* 99.8%

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

   16. Simplified 99.8%

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

   if 2.2000000000000001e-6 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   7. 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. associate-*r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \left(\frac{\varepsilon \cdot \sin x}{\cos x} + \frac{{\varepsilon}^{2}}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0 \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -5.8e-7)
     (fma
      (* t_0 (fma (tan x) (tan eps) 1.0))
      (/ 1.0 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
      (- (tan x)))
     (if (<= eps 1.8e-7)
       (+
        (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ (cos eps) (sin eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -5.8e-7) {
		tmp = fma((t_0 * fma(tan(x), tan(eps), 1.0)), (1.0 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), -tan(x));
	} else if (eps <= 1.8e-7) {
		tmp = (eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0)) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (cos(eps) / sin(eps))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -5.8e-7)
		tmp = fma(Float64(t_0 * fma(tan(x), tan(eps), 1.0)), Float64(1.0 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), Float64(-tan(x)));
	elseif (eps <= 1.8e-7)
		tmp = Float64(Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0)) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(cos(eps) / sin(eps))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.8e-7], N[(N[(t$95$0 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 1.8e-7], N[(N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.7999999999999995e-7

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. associate-/r/ 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow2 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   10. Simplified 99.4%

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

       1. div-inv 99.3%

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

       2. fma-neg 99.5%

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

   12. Applied egg-rr 99.5%

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

   if -5.7999999999999995e-7 < eps < 1.79999999999999997e-7

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 31.2%

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

      4. prod-diff 31.1%

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

      5. *-commutative 31.1%

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

      6. *-un-lft-identity 31.1%

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

      7. *-commutative 31.1%

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

      8. *-un-lft-identity 31.1%

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

   4. Applied egg-rr 31.1%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   if 1.79999999999999997e-7 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   7. 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. associate-*r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0 \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_1}{1 - t_1 \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (/ (sin eps) (cos eps))))
   (if (<= eps -6.4e-9)
     (fma
      (* t_0 (fma (tan x) (tan eps) 1.0))
      (/ 1.0 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
      (- (tan x)))
     (if (<= eps 7.2e-9)
       (+
        (/ t_1 (- 1.0 (* t_1 (/ (sin x) (cos x)))))
        (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0)))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ (cos eps) (sin eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = sin(eps) / cos(eps);
	double tmp;
	if (eps <= -6.4e-9) {
		tmp = fma((t_0 * fma(tan(x), tan(eps), 1.0)), (1.0 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), -tan(x));
	} else if (eps <= 7.2e-9) {
		tmp = (t_1 / (1.0 - (t_1 * (sin(x) / cos(x))))) + ((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (cos(eps) / sin(eps))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if (eps <= -6.4e-9)
		tmp = fma(Float64(t_0 * fma(tan(x), tan(eps), 1.0)), Float64(1.0 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), Float64(-tan(x)));
	elseif (eps <= 7.2e-9)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(t_1 * Float64(sin(x) / cos(x))))) + Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(cos(eps) / sin(eps))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.4e-9], N[(N[(t$95$0 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 7.2e-9], N[(N[(t$95$1 / N[(1.0 - N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.40000000000000023e-9

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. associate-/r/ 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow2 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   10. Simplified 99.4%

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

       1. div-inv 99.3%

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

       2. fma-neg 99.5%

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

   12. Applied egg-rr 99.5%

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

   if -6.40000000000000023e-9 < eps < 7.2e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 31.2%

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

      4. prod-diff 31.1%

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

      5. *-commutative 31.1%

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

      6. *-un-lft-identity 31.1%

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

      7. *-commutative 31.1%

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

      8. *-un-lft-identity 31.1%

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

   4. Applied egg-rr 31.1%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in x around inf 31.1%

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

      1. associate--l+ 58.2%

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

      2. associate-/r* 58.2%

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

      3. times-frac 58.2%

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

   9. Simplified 58.2%

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

   10. Taylor expanded in eps around 0 99.3%

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

   if 7.2e-9 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   7. 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. associate-*r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t_0 \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_1}{1 - t_1 \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (/ (sin eps) (cos eps))))
   (if (<= eps -2.3e-8)
     (fma
      (* t_0 (fma (tan x) (tan eps) 1.0))
      (/ 1.0 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
      (- (tan x)))
     (if (<= eps 6e-9)
       (+
        (/ t_1 (- 1.0 (* t_1 (/ (sin x) (cos x)))))
        (* (tan x) (/ (* eps (sin x)) (cos x))))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ (cos eps) (sin eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = sin(eps) / cos(eps);
	double tmp;
	if (eps <= -2.3e-8) {
		tmp = fma((t_0 * fma(tan(x), tan(eps), 1.0)), (1.0 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), -tan(x));
	} else if (eps <= 6e-9) {
		tmp = (t_1 / (1.0 - (t_1 * (sin(x) / cos(x))))) + (tan(x) * ((eps * sin(x)) / cos(x)));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (cos(eps) / sin(eps))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if (eps <= -2.3e-8)
		tmp = fma(Float64(t_0 * fma(tan(x), tan(eps), 1.0)), Float64(1.0 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), Float64(-tan(x)));
	elseif (eps <= 6e-9)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(t_1 * Float64(sin(x) / cos(x))))) + Float64(tan(x) * Float64(Float64(eps * sin(x)) / cos(x))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(cos(eps) / sin(eps))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.3e-8], N[(N[(t$95$0 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 6e-9], N[(N[(t$95$1 / N[(1.0 - N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.3000000000000001e-8

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. associate-/r/ 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow2 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   10. Simplified 99.4%

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

       1. div-inv 99.3%

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

       2. fma-neg 99.5%

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

   12. Applied egg-rr 99.5%

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

   if -2.3000000000000001e-8 < eps < 5.99999999999999996e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 31.2%

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

      4. prod-diff 31.1%

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

      5. *-commutative 31.1%

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

      6. *-un-lft-identity 31.1%

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

      7. *-commutative 31.1%

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

      8. *-un-lft-identity 31.1%

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

   4. Applied egg-rr 31.1%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in x around inf 31.1%

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

      1. associate--l+ 58.2%

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

      2. associate-/r* 58.2%

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

      3. times-frac 58.2%

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

   9. Simplified 58.2%

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

       1. tan-quot 55.5%

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

       2. tan-quot 58.2%

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

       3. div-inv 58.2%

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

       4. fma-neg 58.2%

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

       5. tan-quot 58.2%

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

       6. tan-quot 58.2%

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

       7. *-commutative 58.2%

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

   11. Applied egg-rr 58.2%

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

       1. fma-udef 58.2%

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

       2. *-rgt-identity 58.2%

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

       3. distribute-rgt-neg-in 58.2%

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

       4. metadata-eval 58.2%

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

       5. distribute-lft-out 58.2%

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

   13. Simplified 58.2%

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

   14. Taylor expanded in eps around 0 99.3%

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

   if 5.99999999999999996e-9 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   7. 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. associate-*r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -4.8e-9)
     (fma (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_1 (- (tan x)))
     (if (<= eps 5.8e-9)
       (+
        (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
        (* (tan x) (/ (* eps (sin x)) (cos x))))
       (- (/ t_1 (- 1.0 (/ (tan x) (/ (cos eps) (sin eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.8e-9) {
		tmp = fma((1.0 / (1.0 - (tan(x) * tan(eps)))), t_1, -tan(x));
	} else if (eps <= 5.8e-9) {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + (tan(x) * ((eps * sin(x)) / cos(x)));
	} else {
		tmp = (t_1 / (1.0 - (tan(x) / (cos(eps) / sin(eps))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.8e-9)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_1, Float64(-tan(x)));
	elseif (eps <= 5.8e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(tan(x) * Float64(Float64(eps * sin(x)) / cos(x))));
	else
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(tan(x) / Float64(cos(eps) / sin(eps))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.8e-9], N[(N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 5.8e-9], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.8e-9

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. div-inv 99.4%

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

      2. *-commutative 99.4%

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

      3. fma-neg 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -4.8e-9 < eps < 5.79999999999999982e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 31.2%

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

      4. prod-diff 31.1%

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

      5. *-commutative 31.1%

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

      6. *-un-lft-identity 31.1%

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

      7. *-commutative 31.1%

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

      8. *-un-lft-identity 31.1%

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

   4. Applied egg-rr 31.1%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in x around inf 31.1%

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

      1. associate--l+ 58.2%

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

      2. associate-/r* 58.2%

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

      3. times-frac 58.2%

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

   9. Simplified 58.2%

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

       1. tan-quot 55.5%

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

       2. tan-quot 58.2%

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

       3. div-inv 58.2%

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

       4. fma-neg 58.2%

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

       5. tan-quot 58.2%

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

       6. tan-quot 58.2%

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

       7. *-commutative 58.2%

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

   11. Applied egg-rr 58.2%

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

       1. fma-udef 58.2%

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

       2. *-rgt-identity 58.2%

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

       3. distribute-rgt-neg-in 58.2%

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

       4. metadata-eval 58.2%

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

       5. distribute-lft-out 58.2%

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

   13. Simplified 58.2%

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

   14. Taylor expanded in eps around 0 99.3%

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

   if 5.79999999999999982e-9 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   7. 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. associate-*r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \tan x \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2.9e-9)
     (fma (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_0 (- (tan x)))
     (if (<= eps 5e-9)
       (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ (cos eps) (sin eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.9e-9) {
		tmp = fma((1.0 / (1.0 - (tan(x) * tan(eps)))), t_0, -tan(x));
	} else if (eps <= 5e-9) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (cos(eps) / sin(eps))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.9e-9)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_0, Float64(-tan(x)));
	elseif (eps <= 5e-9)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(cos(eps) / sin(eps))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.9e-9], N[(N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 5e-9], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.89999999999999991e-9

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. div-inv 99.4%

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

      2. *-commutative 99.4%

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

      3. fma-neg 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -2.89999999999999991e-9 < eps < 5.0000000000000001e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   4. 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%

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

   5. Simplified 99.2%

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

   if 5.0000000000000001e-9 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   7. 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. associate-*r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_0}, t_1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps)))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -3.4e-9)
     (fma (/ 1.0 t_0) t_1 (- (tan x)))
     (if (<= eps 2.6e-9)
       (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
       (- (/ t_1 t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.4e-9) {
		tmp = fma((1.0 / t_0), t_1, -tan(x));
	} else if (eps <= 2.6e-9) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = (t_1 / t_0) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.4e-9)
		tmp = fma(Float64(1.0 / t_0), t_1, Float64(-tan(x)));
	elseif (eps <= 2.6e-9)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(t_1 / t_0) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.4e-9], N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$1 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.6e-9], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.3999999999999998e-9

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

      1. div-inv 99.4%

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

      2. *-commutative 99.4%

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

      3. fma-neg 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -3.3999999999999998e-9 < eps < 2.6000000000000001e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   4. 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%

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

   5. Simplified 99.2%

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

   if 2.6000000000000001e-9 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3e-9) (not (<= eps 2e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-9) || !(eps <= 2e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3d-9)) .or. (.not. (eps <= 2d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-9) || !(eps <= 2e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3e-9) or not (eps <= 2e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3e-9) || !(eps <= 2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3e-9) || ~((eps <= 2e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3e-9], N[Not[LessEqual[eps, 2e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.99999999999999998e-9 or 2.00000000000000012e-9 < eps

   1. Initial program 58.1%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.4%

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

   if -2.99999999999999998e-9 < eps < 2.00000000000000012e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   4. 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%

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

   5. Simplified 99.2%

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

4. Final simplification 99.3%

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

Alternative 9: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -3.8e-9)
     (- (* t_0 (/ 1.0 t_1)) (tan x))
     (if (<= eps 3.5e-9)
       (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
       (- (/ t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -3.8e-9) {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	} else if (eps <= 3.5e-9) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = (t_0 / t_1) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    if (eps <= (-3.8d-9)) then
        tmp = (t_0 * (1.0d0 / t_1)) - tan(x)
    else if (eps <= 3.5d-9) then
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    else
        tmp = (t_0 / t_1) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double tmp;
	if (eps <= -3.8e-9) {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	} else if (eps <= 3.5e-9) {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	} else {
		tmp = (t_0 / t_1) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	tmp = 0
	if eps <= -3.8e-9:
		tmp = (t_0 * (1.0 / t_1)) - math.tan(x)
	elif eps <= 3.5e-9:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	else:
		tmp = (t_0 / t_1) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -3.8e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	elseif (eps <= 3.5e-9)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = 1.0 - (tan(x) * tan(eps));
	tmp = 0.0;
	if (eps <= -3.8e-9)
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	elseif (eps <= 3.5e-9)
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	else
		tmp = (t_0 / t_1) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.8e-9], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.5e-9], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.80000000000000011e-9

   1. Initial program 61.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -3.80000000000000011e-9 < eps < 3.4999999999999999e-9

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   4. 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%

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

   5. Simplified 99.2%

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

   if 3.4999999999999999e-9 < eps

   1. Initial program 54.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-un-lft-identity 99.5%

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

      4. prod-diff 99.5%

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

      5. *-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. *-un-lft-identity 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon} - x} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.25e-6)
   (/ (sin eps) (cos eps))
   (if (<= eps 2.3e-5)
     (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
     (- (/ 1.0 (- (/ (cos eps) (sin eps)) x)) (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.25e-6) {
		tmp = sin(eps) / cos(eps);
	} else if (eps <= 2.3e-5) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = (1.0 / ((cos(eps) / sin(eps)) - x)) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.25d-6)) then
        tmp = sin(eps) / cos(eps)
    else if (eps <= 2.3d-5) then
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    else
        tmp = (1.0d0 / ((cos(eps) / sin(eps)) - x)) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.25e-6) {
		tmp = Math.sin(eps) / Math.cos(eps);
	} else if (eps <= 2.3e-5) {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	} else {
		tmp = (1.0 / ((Math.cos(eps) / Math.sin(eps)) - x)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.25e-6:
		tmp = math.sin(eps) / math.cos(eps)
	elif eps <= 2.3e-5:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	else:
		tmp = (1.0 / ((math.cos(eps) / math.sin(eps)) - x)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.25e-6)
		tmp = Float64(sin(eps) / cos(eps));
	elseif (eps <= 2.3e-5)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(cos(eps) / sin(eps)) - x)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.25e-6)
		tmp = sin(eps) / cos(eps);
	elseif (eps <= 2.3e-5)
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	else
		tmp = (1.0 / ((cos(eps) / sin(eps)) - x)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.25e-6], N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.3e-5], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.2500000000000001e-6

   1. Initial program 61.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.3%

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

   if -1.2500000000000001e-6 < eps < 2.3e-5

   1. Initial program 30.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   4. 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%

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

   5. Simplified 99.2%

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

   if 2.3e-5 < eps

   1. Initial program 54.3%

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

      1. tan-quot 54.2%

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

      2. clear-num 54.1%

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

   4. Applied egg-rr 54.1%

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

   5. Taylor expanded in x around 0 55.7%

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

   6. Taylor expanded in x around 0 56.4%

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

      1. +-commutative 56.4%

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

      2. neg-mul-1 56.4%

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

      3. unsub-neg 56.4%

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

   8. Simplified 56.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon} - x} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 59.2%

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

4. Final simplification 59.2%

   \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon} \]
5. Add Preprocessing

Alternative 12: 39.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 42.9%

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

4. Final simplification 42.9%

   \[\leadsto \tan \left(\varepsilon + x\right) - x \]
5. Add Preprocessing

Alternative 13: 3.6% accurate, 102.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 42.9%

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

4. Taylor expanded in x around inf 3.8%

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

   1. neg-mul-1 3.8%

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

6. Simplified 3.8%

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

7. Final simplification 3.8%

   \[\leadsto -x \]
8. Add Preprocessing

Developer target: 75.9% 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 2024011 
(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)))