2tan (problem 3.3.2)

Percentage Accurate: 41.3% → 99.5%

Time: 19.1s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\\ t_3 := \tan x + \tan \varepsilon\\ t_4 := \frac{t_0}{t_1}\\ t_5 := 1 + t_4\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, t_3, \sin x \cdot t_2\right)}{t_2 \cdot \left(-\cos x\right)}\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot t_5 + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t_4 \cdot 0.3333333333333333 - \frac{t_0 \cdot \left(-1 - t_4\right)}{t_1}\right)\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot t_5\right)}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_3}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (fma (tan x) (tan eps) -1.0))
        (t_3 (+ (tan x) (tan eps)))
        (t_4 (/ t_0 t_1))
        (t_5 (+ 1.0 t_4)))
   (if (<= eps -5e-5)
     (/ (fma (cos x) t_3 (* (sin x) t_2)) (* t_2 (- (cos x))))
     (if (<= eps 5.4e-6)
       (+
        (* eps t_5)
        (+
         (*
          (pow eps 3.0)
          (+
           0.3333333333333333
           (- (* t_4 0.3333333333333333) (/ (* t_0 (- -1.0 t_4)) t_1))))
         (/ (* (pow eps 2.0) (* (sin x) t_5)) (cos x))))
       (- (- (tan x)) (/ t_3 t_2))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = fma(tan(x), tan(eps), -1.0);
	double t_3 = tan(x) + tan(eps);
	double t_4 = t_0 / t_1;
	double t_5 = 1.0 + t_4;
	double tmp;
	if (eps <= -5e-5) {
		tmp = fma(cos(x), t_3, (sin(x) * t_2)) / (t_2 * -cos(x));
	} else if (eps <= 5.4e-6) {
		tmp = (eps * t_5) + ((pow(eps, 3.0) * (0.3333333333333333 + ((t_4 * 0.3333333333333333) - ((t_0 * (-1.0 - t_4)) / t_1)))) + ((pow(eps, 2.0) * (sin(x) * t_5)) / cos(x)));
	} else {
		tmp = -tan(x) - (t_3 / t_2);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = fma(tan(x), tan(eps), -1.0)
	t_3 = Float64(tan(x) + tan(eps))
	t_4 = Float64(t_0 / t_1)
	t_5 = Float64(1.0 + t_4)
	tmp = 0.0
	if (eps <= -5e-5)
		tmp = Float64(fma(cos(x), t_3, Float64(sin(x) * t_2)) / Float64(t_2 * Float64(-cos(x))));
	elseif (eps <= 5.4e-6)
		tmp = Float64(Float64(eps * t_5) + Float64(Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(Float64(t_4 * 0.3333333333333333) - Float64(Float64(t_0 * Float64(-1.0 - t_4)) / t_1)))) + Float64(Float64((eps ^ 2.0) * Float64(sin(x) * t_5)) / cos(x))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_3 / t_2));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 + t$95$4), $MachinePrecision]}, If[LessEqual[eps, -5e-5], N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + N[(N[Sin[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * (-N[Cos[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.4e-6], N[(N[(eps * t$95$5), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(t$95$4 * 0.3333333333333333), $MachinePrecision] - N[(N[(t$95$0 * N[(-1.0 - t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.00000000000000024e-5

   1. Initial program 59.1%

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

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   3. Applied egg-rr 28.3%

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

      1. unpow2 28.3%

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

      2. add-sqr-sqrt 59.1%

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

      3. tan-sum 99.5%

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

      4. div-inv 99.6%

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

      5. frac-2neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. associate-*r/ 99.5%

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

      8. sub-neg 99.5%

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

      9. distribute-neg-in 99.5%

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

      10. metadata-eval 99.5%

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

      11. distribute-lft-neg-in 99.5%

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

      12. add-sqr-sqrt 38.2%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. tan-quot 99.6%

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

      3. frac-sub 99.6%

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

      4. remove-double-neg 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. fma-neg 99.6%

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

       3. distribute-lft-neg-out 99.6%

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

       4. remove-double-neg 99.6%

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

       5. *-commutative 99.6%

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

       6. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   if -5.00000000000000024e-5 < eps < 5.39999999999999997e-6

   1. Initial program 29.9%

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

      1. add-sqr-sqrt 15.3%

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

      2. pow2 15.3%

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

   3. Applied egg-rr 15.3%

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

      1. unpow2 15.3%

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

      2. add-sqr-sqrt 29.9%

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

      3. tan-sum 31.0%

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

      4. div-inv 31.0%

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

      5. frac-2neg 31.0%

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

      6. metadata-eval 31.0%

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

      7. associate-*r/ 31.0%

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

      8. sub-neg 31.0%

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

      9. distribute-neg-in 31.0%

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

      10. metadata-eval 31.0%

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

      11. distribute-lft-neg-in 31.0%

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

      12. add-sqr-sqrt 16.3%

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

   5. Applied egg-rr 31.0%

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

      1. *-commutative 31.0%

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

      2. neg-mul-1 31.0%

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

      3. +-commutative 31.0%

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

      4. fma-def 31.0%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]

   if 5.39999999999999997e-6 < eps

   1. Initial program 46.6%

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

      1. add-sqr-sqrt 21.1%

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

      2. pow2 21.1%

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

   3. Applied egg-rr 21.1%

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

      1. unpow2 21.1%

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

      2. add-sqr-sqrt 46.6%

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

      3. tan-sum 99.5%

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

      4. div-inv 99.5%

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

      5. frac-2neg 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-*r/ 99.5%

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

      8. sub-neg 99.5%

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

      9. distribute-neg-in 99.5%

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

      10. metadata-eval 99.5%

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

      11. distribute-lft-neg-in 99.5%

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

      12. add-sqr-sqrt 56.2%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \sin x \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right) \cdot \left(-\cos x\right)}\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.3333333333333333 - \frac{{\sin x}^{2} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \sin x \cdot t_0\right)}{t_0 \cdot \left(-\cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (tan x) (tan eps) -1.0)))
   (if (or (<= eps -1.45e-7) (not (<= eps 3.9e-19)))
     (/
      (fma (cos x) (+ (tan x) (tan eps)) (* (sin x) t_0))
      (* t_0 (- (cos x))))
     (+
      (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
      (*
       (pow eps 2.0)
       (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x))))))))

C

double code(double x, double eps) {
	double t_0 = fma(tan(x), tan(eps), -1.0);
	double tmp;
	if ((eps <= -1.45e-7) || !(eps <= 3.9e-19)) {
		tmp = fma(cos(x), (tan(x) + tan(eps)), (sin(x) * t_0)) / (t_0 * -cos(x));
	} else {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(tan(x), tan(eps), -1.0)
	tmp = 0.0
	if ((eps <= -1.45e-7) || !(eps <= 3.9e-19))
		tmp = Float64(fma(cos(x), Float64(tan(x) + tan(eps)), Float64(sin(x) * t_0)) / Float64(t_0 * Float64(-cos(x))));
	else
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[eps, -1.45e-7], N[Not[LessEqual[eps, 3.9e-19]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * (-N[Cos[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.4499999999999999e-7 or 3.89999999999999995e-19 < eps

   1. Initial program 53.9%

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

      1. add-sqr-sqrt 25.3%

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

      2. pow2 25.3%

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

   3. Applied egg-rr 25.3%

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

      1. unpow2 25.3%

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

      2. add-sqr-sqrt 53.9%

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

      3. tan-sum 99.3%

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

      4. div-inv 99.3%

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

      5. frac-2neg 99.3%

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

      6. metadata-eval 99.3%

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

      7. associate-*r/ 99.3%

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

      8. sub-neg 99.3%

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

      9. distribute-neg-in 99.3%

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

      10. metadata-eval 99.3%

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

      11. distribute-lft-neg-in 99.3%

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

      12. add-sqr-sqrt 46.4%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.4%

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

   7. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. tan-quot 99.4%

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

      3. frac-sub 99.3%

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

      4. remove-double-neg 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. *-commutative 99.3%

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

       2. fma-neg 99.5%

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

       3. distribute-lft-neg-out 99.5%

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

       4. remove-double-neg 99.5%

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

       5. *-commutative 99.5%

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

       6. *-commutative 99.5%

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

   11. Simplified 99.5%

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

   if -1.4499999999999999e-7 < eps < 3.89999999999999995e-19

   1. Initial program 28.9%

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

      1. tan-sum 29.4%

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

      2. div-inv 29.4%

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

      3. tan-quot 29.3%

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

      4. div-inv 29.1%

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

      5. prod-diff 28.7%

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

   3. Applied egg-rr 28.7%

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

      1. +-commutative 28.7%

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

      2. fma-udef 28.7%

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

      3. associate-+r+ 29.1%

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

      4. unsub-neg 29.1%

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

   5. Simplified 29.3%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      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%

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

   8. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \sin x \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right) \cdot \left(-\cos x\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.35e-7) (not (<= eps 2e-6)))
   (- (- (tan x)) (/ (+ (tan x) (tan eps)) (fma (tan x) (tan eps) -1.0)))
   (+
    (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
    (*
     (pow eps 2.0)
     (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.35e-7) || !(eps <= 2e-6)) {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0));
	} else {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.35e-7) || !(eps <= 2e-6))
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0)));
	else
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.35e-7], N[Not[LessEqual[eps, 2e-6]], $MachinePrecision]], N[((-N[Tan[x], $MachinePrecision]) - N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.35000000000000004e-7 or 1.99999999999999991e-6 < eps

   1. Initial program 53.9%

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

      1. add-sqr-sqrt 24.9%

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

      2. pow2 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. unpow2 24.9%

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

      2. add-sqr-sqrt 53.9%

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

      3. tan-sum 99.5%

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

      4. div-inv 99.5%

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

      5. frac-2neg 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-*r/ 99.5%

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

      8. sub-neg 99.5%

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

      9. distribute-neg-in 99.5%

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

      10. metadata-eval 99.5%

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

      11. distribute-lft-neg-in 99.5%

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

      12. add-sqr-sqrt 45.9%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

   if -1.35000000000000004e-7 < eps < 1.99999999999999991e-6

   1. Initial program 29.3%

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

      1. tan-sum 30.4%

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

      2. div-inv 30.4%

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

      3. tan-quot 30.3%

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

      4. div-inv 30.1%

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

      5. prod-diff 29.8%

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

   3. Applied egg-rr 29.8%

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

      1. +-commutative 29.8%

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

      2. fma-udef 29.7%

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

      3. associate-+r+ 30.1%

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

      4. unsub-neg 30.1%

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

   5. Simplified 30.3%

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

   6. Taylor expanded in eps around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. cancel-sign-sub-inv 99.4%

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

      5. metadata-eval 99.4%

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

      6. *-lft-identity 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.5%

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

Alternative 4: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - t_1\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_2}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_0, -\cos x, \sin x \cdot t_2\right)}{\cos x \cdot \left(-1 + t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps)))
        (t_1 (* (tan x) (tan eps)))
        (t_2 (- 1.0 t_1)))
   (if (<= eps -3.1e-9)
     (fma t_0 (/ 1.0 t_2) (- (tan x)))
     (if (<= eps 3.9e-19)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (/ (fma t_0 (- (cos x)) (* (sin x) t_2)) (* (cos x) (+ -1.0 t_1)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = tan(x) * tan(eps);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (eps <= -3.1e-9) {
		tmp = fma(t_0, (1.0 / t_2), -tan(x));
	} else if (eps <= 3.9e-19) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = fma(t_0, -cos(x), (sin(x) * t_2)) / (cos(x) * (-1.0 + t_1));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(tan(x) * tan(eps))
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (eps <= -3.1e-9)
		tmp = fma(t_0, Float64(1.0 / t_2), Float64(-tan(x)));
	elseif (eps <= 3.9e-19)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(fma(t_0, Float64(-cos(x)), Float64(sin(x) * t_2)) / Float64(cos(x) * Float64(-1.0 + t_1)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[eps, -3.1e-9], N[(t$95$0 * N[(1.0 / t$95$2), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 3.9e-19], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * (-N[Cos[x], $MachinePrecision]) + N[(N[Sin[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.10000000000000005e-9

   1. Initial program 58.9%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.1%

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

      3. tan-quot 99.1%

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

      4. div-inv 99.0%

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

      5. prod-diff 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-udef 99.0%

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

      3. associate-+r+ 99.0%

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

      4. unsub-neg 99.0%

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

   5. Simplified 99.0%

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

      1. tan-quot 99.0%

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

      2. expm1-log1p-u 86.7%

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

      3. expm1-udef 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. div-inv 86.6%

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

      2. expm1-def 86.7%

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

      3. expm1-log1p-u 99.1%

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

      4. fma-neg 99.1%

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

   9. Applied egg-rr 99.1%

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

   if -3.10000000000000005e-9 < eps < 3.89999999999999995e-19

   1. Initial program 29.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   4. Simplified 99.6%

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

   if 3.89999999999999995e-19 < eps

   1. Initial program 46.7%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.0%

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

      3. tan-quot 99.0%

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

      4. div-inv 98.9%

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

      5. prod-diff 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-udef 98.9%

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

      3. associate-+r+ 99.0%

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

      4. unsub-neg 99.0%

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

   5. Simplified 99.1%

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

      1. expm1-log1p-u 85.1%

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

      2. expm1-udef 85.0%

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

      3. log1p-udef 85.0%

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

      4. add-exp-log 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. frac-2neg 99.0%

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

      2. frac-sub 98.9%

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

      3. add-exp-log 84.9%

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

      4. log1p-udef 84.9%

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

      5. expm1-udef 85.0%

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

      6. expm1-log1p-u 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. fma-neg 99.2%

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

       2. distribute-rgt-neg-out 99.2%

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

       3. remove-double-neg 99.2%

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

       4. *-commutative 99.2%

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

       5. distribute-rgt-neg-in 99.2%

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

       6. distribute-lft-neg-in 99.2%

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

       7. *-commutative 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\tan x + \tan \varepsilon, -\cos x, \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(-1 + \tan x \cdot \tan \varepsilon\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-19}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.9e-9) (not (<= eps 3.9e-19)))
   (- (- (tan x)) (/ (+ (tan x) (tan eps)) (fma (tan x) (tan eps) -1.0)))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.9e-9) || !(eps <= 3.9e-19)) {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0));
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.9e-9) || !(eps <= 3.9e-19))
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0)));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.9e-9], N[Not[LessEqual[eps, 3.9e-19]], $MachinePrecision]], N[((-N[Tan[x], $MachinePrecision]) - N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.89999999999999991e-9 or 3.89999999999999995e-19 < eps

   1. Initial program 53.5%

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

      1. add-sqr-sqrt 25.2%

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

      2. pow2 25.2%

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

   3. Applied egg-rr 25.2%

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

      1. unpow2 25.2%

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

      2. add-sqr-sqrt 53.5%

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

      3. tan-sum 99.1%

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

      4. div-inv 99.1%

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

      5. frac-2neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. associate-*r/ 99.1%

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

      8. sub-neg 99.1%

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

      9. distribute-neg-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. distribute-lft-neg-in 99.1%

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

      12. add-sqr-sqrt 46.1%

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

   5. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. neg-mul-1 99.1%

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

      3. +-commutative 99.1%

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

      4. fma-def 99.2%

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

   7. Simplified 99.2%

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

   if -2.89999999999999991e-9 < eps < 3.89999999999999995e-19

   1. Initial program 29.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-19}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- (tan x))))
   (if (<= eps -3.2e-9)
     (fma t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_1)
     (if (<= eps 3.9e-19)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- t_1 (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = -tan(x);
	double tmp;
	if (eps <= -3.2e-9) {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_1);
	} else if (eps <= 3.9e-19) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = t_1 - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(-tan(x))
	tmp = 0.0
	if (eps <= -3.2e-9)
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_1);
	elseif (eps <= 3.9e-19)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(t_1 - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[x], $MachinePrecision])}, If[LessEqual[eps, -3.2e-9], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 3.9e-19], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.20000000000000012e-9

   1. Initial program 58.9%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.1%

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

      3. tan-quot 99.1%

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

      4. div-inv 99.0%

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

      5. prod-diff 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-udef 99.0%

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

      3. associate-+r+ 99.0%

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

      4. unsub-neg 99.0%

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

   5. Simplified 99.0%

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

      1. tan-quot 99.0%

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

      2. expm1-log1p-u 86.7%

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

      3. expm1-udef 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. div-inv 86.6%

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

      2. expm1-def 86.7%

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

      3. expm1-log1p-u 99.1%

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

      4. fma-neg 99.1%

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

   9. Applied egg-rr 99.1%

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

   if -3.20000000000000012e-9 < eps < 3.89999999999999995e-19

   1. Initial program 29.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   4. Simplified 99.6%

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

   if 3.89999999999999995e-19 < eps

   1. Initial program 46.7%

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

      1. add-sqr-sqrt 22.0%

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

      2. pow2 22.0%

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

   3. Applied egg-rr 22.0%

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

      1. unpow2 22.0%

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

      2. add-sqr-sqrt 46.7%

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

      3. tan-sum 99.1%

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

      4. div-inv 99.0%

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

      5. frac-2neg 99.0%

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

      6. metadata-eval 99.0%

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

      7. associate-*r/ 99.1%

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

      8. sub-neg 99.1%

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

      9. distribute-neg-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. distribute-lft-neg-in 99.1%

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

      12. add-sqr-sqrt 57.2%

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

   5. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. neg-mul-1 99.1%

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

      3. +-commutative 99.1%

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

      4. fma-def 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 7: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7e-9) (not (<= eps 3.9e-19)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-9) || !(eps <= 3.9e-19)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.7d-9)) .or. (.not. (eps <= 3.9d-19))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-9) || !(eps <= 3.9e-19)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.7e-9) or not (eps <= 3.9e-19):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.7e-9) || !(eps <= 3.9e-19))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.7e-9) || ~((eps <= 3.9e-19)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.7e-9], N[Not[LessEqual[eps, 3.9e-19]], $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[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.6999999999999999e-9 or 3.89999999999999995e-19 < eps

   1. Initial program 53.5%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-udef 99.1%

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

      3. associate-+r+ 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

   if -1.6999999999999999e-9 < eps < 3.89999999999999995e-19

   1. Initial program 29.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 8: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -3e-9)
     (- (* t_0 (/ 1.0 t_1)) (tan x))
     (if (<= eps 3.9e-19)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (/ 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 <= -3e-9) {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	} else if (eps <= 3.9e-19) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} 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 <= (-3d-9)) then
        tmp = (t_0 * (1.0d0 / t_1)) - tan(x)
    else if (eps <= 3.9d-19) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    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 <= -3e-9) {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	} else if (eps <= 3.9e-19) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} 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 <= -3e-9:
		tmp = (t_0 * (1.0 / t_1)) - math.tan(x)
	elif eps <= 3.9e-19:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	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 <= -3e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	elseif (eps <= 3.9e-19)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	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 <= -3e-9)
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	elseif (eps <= 3.9e-19)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	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, -3e-9], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.9e-19], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.99999999999999998e-9

   1. Initial program 58.9%

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

      1. tan-sum 99.0%

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

      2. div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

   if -2.99999999999999998e-9 < eps < 3.89999999999999995e-19

   1. Initial program 29.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   4. Simplified 99.6%

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

   if 3.89999999999999995e-19 < eps

   1. Initial program 46.7%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. prod-diff 99.0%

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

      5. *-commutative 99.0%

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

      6. *-un-lft-identity 99.0%

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

      7. *-commutative 99.0%

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

      8. *-un-lft-identity 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-udef 99.0%

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

      3. associate-+r+ 99.0%

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

      4. unsub-neg 99.0%

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

   5. Simplified 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 9: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.45e-7)
   (tan eps)
   (if (<= eps 4e-6)
     (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (- (- (tan x)) (/ (+ (tan x) (tan eps)) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.45e-7) {
		tmp = tan(eps);
	} else if (eps <= 4e-6) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / -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 <= (-1.45d-7)) then
        tmp = tan(eps)
    else if (eps <= 4d-6) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = -tan(x) - ((tan(x) + tan(eps)) / (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.45e-7) {
		tmp = Math.tan(eps);
	} else if (eps <= 4e-6) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = -Math.tan(x) - ((Math.tan(x) + Math.tan(eps)) / -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.45e-7:
		tmp = math.tan(eps)
	elif eps <= 4e-6:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = -math.tan(x) - ((math.tan(x) + math.tan(eps)) / -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.45e-7)
		tmp = tan(eps);
	elseif (eps <= 4e-6)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.45e-7)
		tmp = tan(eps);
	elseif (eps <= 4e-6)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = -tan(x) - ((tan(x) + tan(eps)) / -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.45e-7], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 4e-6], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.4499999999999999e-7

   1. Initial program 59.6%

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

   2. Taylor expanded in x around 0 63.0%

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

      1. tan-quot 63.3%

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

      2. expm1-log1p-u 49.9%

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

      3. expm1-udef 48.9%

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

   4. Applied egg-rr 48.9%

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

      1. expm1-def 49.9%

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

      2. expm1-log1p 63.3%

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

   6. Simplified 63.3%

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

   if -1.4499999999999999e-7 < eps < 3.99999999999999982e-6

   1. Initial program 29.3%

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

   2. Taylor expanded in eps around 0 98.8%

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

      1. cancel-sign-sub-inv 98.8%

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

      2. metadata-eval 98.8%

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

      3. *-lft-identity 98.8%

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

   4. Simplified 98.8%

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

   if 3.99999999999999982e-6 < eps

   1. Initial program 46.6%

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

      1. add-sqr-sqrt 21.1%

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

      2. pow2 21.1%

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

   3. Applied egg-rr 21.1%

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

      1. unpow2 21.1%

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

      2. add-sqr-sqrt 46.6%

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

      3. tan-sum 99.5%

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

      4. div-inv 99.5%

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

      5. frac-2neg 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-*r/ 99.5%

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

      8. sub-neg 99.5%

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

      9. distribute-neg-in 99.5%

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

      10. metadata-eval 99.5%

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

      11. distribute-lft-neg-in 99.5%

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

      12. add-sqr-sqrt 56.2%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 49.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{-1}\\ \end{array} \]

Alternative 10: 57.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

2. Taylor expanded in x around 0 59.4%

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

   1. tan-quot 59.5%

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

   2. expm1-log1p-u 51.3%

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

   3. expm1-udef 25.5%

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

4. Applied egg-rr 25.5%

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

   1. expm1-def 51.3%

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

   2. expm1-log1p 59.5%

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

6. Simplified 59.5%

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

7. Final simplification 59.5%

   \[\leadsto \tan \varepsilon \]

Alternative 11: 31.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 42.8%

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

2. Taylor expanded in x around 0 59.4%

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

3. Taylor expanded in eps around 0 30.6%

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

4. Final simplification 30.6%

   \[\leadsto \varepsilon \]

Developer target: 76.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023333 
(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)))