2tan (problem 3.3.2)

Percentage Accurate: 42.2% → 99.5%

Time: 17.4s

Alternatives: 13

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\\ t_2 := -\tan x\\ t_3 := \tan x + \tan \varepsilon\\ t_4 := \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ t_5 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_6 := t_5 + 0.3333333333333333\\ t_7 := t_5 \cdot -0.3333333333333333\\ \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\mathsf{fma}\left(t_3, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot t_0 + 1}{1 - {t_0}^{3}}, t_2\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000195:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_5 + 1, \mathsf{fma}\left({\varepsilon}^{3}, t_6 + \left(t_4 - t_7\right), t_1 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{t_6}{\frac{\cos x}{\sin x}} - \mathsf{fma}\left(-0.3333333333333333, t_1, \frac{\sin x}{\frac{\cos x}{t_7 - t_4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_3, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps)))
        (t_1 (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x))))
        (t_2 (- (tan x)))
        (t_3 (+ (tan x) (tan eps)))
        (t_4 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
        (t_5 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_6 (+ t_5 0.3333333333333333))
        (t_7 (* t_5 -0.3333333333333333)))
   (if (<= eps -0.00023)
     (fma
      t_3
      (/ (+ (* (fma (tan x) (tan eps) 1.0) t_0) 1.0) (- 1.0 (pow t_0 3.0)))
      t_2)
     (if (<= eps 0.000195)
       (+
        (fma
         eps
         (+ t_5 1.0)
         (fma (pow eps 3.0) (+ t_6 (- t_4 t_7)) (* t_1 (* eps eps))))
        (*
         (pow eps 4.0)
         (-
          (/ t_6 (/ (cos x) (sin x)))
          (fma -0.3333333333333333 t_1 (/ (sin x) (/ (cos x) (- t_7 t_4)))))))
       (fma t_3 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_2)))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = (pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x));
	double t_2 = -tan(x);
	double t_3 = tan(x) + tan(eps);
	double t_4 = pow(sin(x), 4.0) / pow(cos(x), 4.0);
	double t_5 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_6 = t_5 + 0.3333333333333333;
	double t_7 = t_5 * -0.3333333333333333;
	double tmp;
	if (eps <= -0.00023) {
		tmp = fma(t_3, (((fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / (1.0 - pow(t_0, 3.0))), t_2);
	} else if (eps <= 0.000195) {
		tmp = fma(eps, (t_5 + 1.0), fma(pow(eps, 3.0), (t_6 + (t_4 - t_7)), (t_1 * (eps * eps)))) + (pow(eps, 4.0) * ((t_6 / (cos(x) / sin(x))) - fma(-0.3333333333333333, t_1, (sin(x) / (cos(x) / (t_7 - t_4))))));
	} else {
		tmp = fma(t_3, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_2);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x)))
	t_2 = Float64(-tan(x))
	t_3 = Float64(tan(x) + tan(eps))
	t_4 = Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))
	t_5 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_6 = Float64(t_5 + 0.3333333333333333)
	t_7 = Float64(t_5 * -0.3333333333333333)
	tmp = 0.0
	if (eps <= -0.00023)
		tmp = fma(t_3, Float64(Float64(Float64(fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / Float64(1.0 - (t_0 ^ 3.0))), t_2);
	elseif (eps <= 0.000195)
		tmp = Float64(fma(eps, Float64(t_5 + 1.0), fma((eps ^ 3.0), Float64(t_6 + Float64(t_4 - t_7)), Float64(t_1 * Float64(eps * eps)))) + Float64((eps ^ 4.0) * Float64(Float64(t_6 / Float64(cos(x) / sin(x))) - fma(-0.3333333333333333, t_1, Float64(sin(x) / Float64(cos(x) / Float64(t_7 - t_4)))))));
	else
		tmp = fma(t_3, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_2);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(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]}, Block[{t$95$2 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$3 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + 0.3333333333333333), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 * -0.3333333333333333), $MachinePrecision]}, If[LessEqual[eps, -0.00023], N[(t$95$3 * N[(N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[eps, 0.000195], N[(N[(eps * N[(t$95$5 + 1.0), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(t$95$6 + N[(t$95$4 - t$95$7), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(t$95$6 / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * t$95$1 + N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(t$95$7 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.3000000000000001e-4

   1. Initial program 46.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. flip3-- 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}}, -\tan x\right) \]

      2. associate-/r/ 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}, -\tan x\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{1} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      4. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(\color{blue}{1} + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      5. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \color{blue}{\tan x \cdot \tan \varepsilon}\right)\right), -\tan x\right) \]

      6. +-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}\right), -\tan x\right) \]

      7. pow2 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}\right)\right), -\tan x\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}, -\tan x\right) \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

      2. *-lft-identity 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      3. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      4. distribute-rgt1-in 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      5. fma-def 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)} \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 + \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

   if -2.3000000000000001e-4 < eps < 1.94999999999999996e-4

   1. Initial program 30.1%

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

      1. tan-sum 32.2%

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

      2. div-inv 32.2%

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

   3. Applied egg-rr 32.2%

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

   4. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{4} \cdot \left(-1 \cdot \frac{\left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x} + \left(\frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(-1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left({\varepsilon}^{3}, \left(0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \left(-0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \left(\frac{-\sin x}{\cos x} - \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot \left(\left(-\varepsilon\right) \cdot \varepsilon\right)\right)\right) - {\varepsilon}^{4} \cdot \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \frac{\sin x}{\frac{\cos x}{-0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}}}\right) - \frac{0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\frac{\cos x}{\sin x}}\right)} \]

   if 1.94999999999999996e-4 < eps

   1. Initial program 44.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. div-inv 99.3%

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

      4. sub-neg 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. distribute-lft-neg-in 99.3%

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

      8. add-sqr-sqrt 39.5%

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

      9. sqrt-unprod 68.9%

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

      10. sqr-neg 68.9%

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

      11. sqrt-unprod 29.4%

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

      12. add-sqr-sqrt 48.0%

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

      13. distribute-lft-neg-in 48.0%

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

      14. add-sqr-sqrt 18.6%

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

      15. sqrt-unprod 78.4%

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

      16. sqr-neg 78.4%

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

      17. sqrt-unprod 59.7%

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

   5. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000195:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1, \mathsf{fma}\left({\varepsilon}^{3}, \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 0.3333333333333333\right) + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333\right), \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\frac{{\sin x}^{2}}{{\cos x}^{2}} + 0.3333333333333333}{\frac{\cos x}{\sin x}} - \mathsf{fma}\left(-0.3333333333333333, \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}, \frac{\sin x}{\frac{\cos x}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := \frac{\sin x}{\cos x}\\ t_3 := -\tan x\\ t_4 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ t_5 := \tan x + \tan \varepsilon\\ t_6 := -0.3333333333333333 \cdot t_2\\ t_7 := t_4 - t_6\\ \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\mathsf{fma}\left(t_5, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot t_0 + 1}{1 - {t_0}^{3}}, t_3\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00023:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(0.3333333333333333 \cdot t_2 + \left(t_7 - \frac{\sin x \cdot \left(t_1 \cdot -0.3333333333333333 + \frac{\sin x \cdot \left(t_6 - t_4\right)}{\cos x}\right)}{\cos x}\right)\right) + \left(\left(t_4 + t_2\right) \cdot {\varepsilon}^{2} + \left(\varepsilon \cdot \left(t_1 + 1\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t_1 + \frac{\sin x \cdot t_7}{\cos x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_5, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps)))
        (t_1 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_2 (/ (sin x) (cos x)))
        (t_3 (- (tan x)))
        (t_4 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (t_5 (+ (tan x) (tan eps)))
        (t_6 (* -0.3333333333333333 t_2))
        (t_7 (- t_4 t_6)))
   (if (<= eps -0.00023)
     (fma
      t_5
      (/ (+ (* (fma (tan x) (tan eps) 1.0) t_0) 1.0) (- 1.0 (pow t_0 3.0)))
      t_3)
     (if (<= eps 0.00023)
       (+
        (*
         (pow eps 4.0)
         (+
          (* 0.3333333333333333 t_2)
          (-
           t_7
           (/
            (*
             (sin x)
             (+
              (* t_1 -0.3333333333333333)
              (/ (* (sin x) (- t_6 t_4)) (cos x))))
            (cos x)))))
        (+
         (* (+ t_4 t_2) (pow eps 2.0))
         (+
          (* eps (+ t_1 1.0))
          (*
           (pow eps 3.0)
           (+ 0.3333333333333333 (+ t_1 (/ (* (sin x) t_7) (cos x))))))))
       (fma t_5 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_3)))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_2 = sin(x) / cos(x);
	double t_3 = -tan(x);
	double t_4 = pow(sin(x), 3.0) / pow(cos(x), 3.0);
	double t_5 = tan(x) + tan(eps);
	double t_6 = -0.3333333333333333 * t_2;
	double t_7 = t_4 - t_6;
	double tmp;
	if (eps <= -0.00023) {
		tmp = fma(t_5, (((fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / (1.0 - pow(t_0, 3.0))), t_3);
	} else if (eps <= 0.00023) {
		tmp = (pow(eps, 4.0) * ((0.3333333333333333 * t_2) + (t_7 - ((sin(x) * ((t_1 * -0.3333333333333333) + ((sin(x) * (t_6 - t_4)) / cos(x)))) / cos(x))))) + (((t_4 + t_2) * pow(eps, 2.0)) + ((eps * (t_1 + 1.0)) + (pow(eps, 3.0) * (0.3333333333333333 + (t_1 + ((sin(x) * t_7) / cos(x)))))));
	} else {
		tmp = fma(t_5, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_3);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_2 = Float64(sin(x) / cos(x))
	t_3 = Float64(-tan(x))
	t_4 = Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))
	t_5 = Float64(tan(x) + tan(eps))
	t_6 = Float64(-0.3333333333333333 * t_2)
	t_7 = Float64(t_4 - t_6)
	tmp = 0.0
	if (eps <= -0.00023)
		tmp = fma(t_5, Float64(Float64(Float64(fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / Float64(1.0 - (t_0 ^ 3.0))), t_3);
	elseif (eps <= 0.00023)
		tmp = Float64(Float64((eps ^ 4.0) * Float64(Float64(0.3333333333333333 * t_2) + Float64(t_7 - Float64(Float64(sin(x) * Float64(Float64(t_1 * -0.3333333333333333) + Float64(Float64(sin(x) * Float64(t_6 - t_4)) / cos(x)))) / cos(x))))) + Float64(Float64(Float64(t_4 + t_2) * (eps ^ 2.0)) + Float64(Float64(eps * Float64(t_1 + 1.0)) + Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(t_1 + Float64(Float64(sin(x) * t_7) / cos(x))))))));
	else
		tmp = fma(t_5, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_3);
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$4 = N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(-0.3333333333333333 * t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$4 - t$95$6), $MachinePrecision]}, If[LessEqual[eps, -0.00023], N[(t$95$5 * N[(N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[eps, 0.00023], N[(N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(0.3333333333333333 * t$95$2), $MachinePrecision] + N[(t$95$7 - N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$1 * -0.3333333333333333), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$6 - t$95$4), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$4 + t$95$2), $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$1 + N[(N[(N[Sin[x], $MachinePrecision] * t$95$7), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.3000000000000001e-4

   1. Initial program 46.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. flip3-- 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}}, -\tan x\right) \]

      2. associate-/r/ 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}, -\tan x\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{1} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      4. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(\color{blue}{1} + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      5. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \color{blue}{\tan x \cdot \tan \varepsilon}\right)\right), -\tan x\right) \]

      6. +-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}\right), -\tan x\right) \]

      7. pow2 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}\right)\right), -\tan x\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}, -\tan x\right) \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

      2. *-lft-identity 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      3. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      4. distribute-rgt1-in 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      5. fma-def 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)} \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 + \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

   if -2.3000000000000001e-4 < eps < 2.3000000000000001e-4

   1. Initial program 30.1%

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

      1. tan-sum 32.2%

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

      2. div-inv 32.2%

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

      3. fma-neg 32.2%

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

   3. Applied egg-rr 32.2%

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

   4. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-1 \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) + -1 \cdot \frac{\left(\frac{\left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) \cdot \sin x}{\cos x} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right)\right) + \left(\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2} + \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(-1 \cdot \frac{\left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) \cdot \sin x}{\cos x} + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

   if 2.3000000000000001e-4 < eps

   1. Initial program 44.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. div-inv 99.3%

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

      4. sub-neg 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. distribute-lft-neg-in 99.3%

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

      8. add-sqr-sqrt 39.5%

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

      9. sqrt-unprod 68.9%

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

      10. sqr-neg 68.9%

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

      11. sqrt-unprod 29.4%

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

      12. add-sqr-sqrt 48.0%

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

      13. distribute-lft-neg-in 48.0%

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

      14. add-sqr-sqrt 18.6%

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

      15. sqrt-unprod 78.4%

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

      16. sqr-neg 78.4%

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

      17. sqrt-unprod 59.7%

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

   5. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00023:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(\left(\frac{{\sin x}^{3}}{{\cos x}^{3}} - -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) - \frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333 + \frac{\sin x \cdot \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} - \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)}{\cos x}\right)}{\cos x}\right)\right) + \left(\left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right) \cdot {\varepsilon}^{2} + \left(\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{\sin x \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} - -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right)}{\cos x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ t_3 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot t_0 + 1}{1 - {t_0}^{3}}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \left(\frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}} - t_3\right)\right), \left(\varepsilon + \varepsilon \cdot t_3\right) + \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps)))
        (t_1 (- (tan x)))
        (t_2 (+ (tan x) (tan eps)))
        (t_3 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (if (<= eps -5.2e-5)
     (fma
      t_2
      (/ (+ (* (fma (tan x) (tan eps) 1.0) t_0) 1.0) (- 1.0 (pow t_0 3.0)))
      t_1)
     (if (<= eps 5e-5)
       (fma
        (pow eps 3.0)
        (+
         0.3333333333333333
         (-
          (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
          (-
           (/ (sin x) (/ (cos x) (/ (* (sin x) -0.3333333333333333) (cos x))))
           t_3)))
        (+
         (+ eps (* eps t_3))
         (*
          (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))
          (* eps eps))))
       (fma t_2 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_1)))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = -tan(x);
	double t_2 = tan(x) + tan(eps);
	double t_3 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double tmp;
	if (eps <= -5.2e-5) {
		tmp = fma(t_2, (((fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / (1.0 - pow(t_0, 3.0))), t_1);
	} else if (eps <= 5e-5) {
		tmp = fma(pow(eps, 3.0), (0.3333333333333333 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - ((sin(x) / (cos(x) / ((sin(x) * -0.3333333333333333) / cos(x)))) - t_3))), ((eps + (eps * t_3)) + (((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))) * (eps * eps))));
	} else {
		tmp = fma(t_2, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(-tan(x))
	t_2 = Float64(tan(x) + tan(eps))
	t_3 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	tmp = 0.0
	if (eps <= -5.2e-5)
		tmp = fma(t_2, Float64(Float64(Float64(fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / Float64(1.0 - (t_0 ^ 3.0))), t_1);
	elseif (eps <= 5e-5)
		tmp = fma((eps ^ 3.0), Float64(0.3333333333333333 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(Float64(sin(x) / Float64(cos(x) / Float64(Float64(sin(x) * -0.3333333333333333) / cos(x)))) - t_3))), Float64(Float64(eps + Float64(eps * t_3)) + Float64(Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x))) * Float64(eps * eps))));
	else
		tmp = fma(t_2, Float64(-1.0 / fma(tan(x), tan(eps), -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[Tan[x], $MachinePrecision])}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.2e-5], N[(t$95$2 * N[(N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 5e-5], N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps + N[(eps * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(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] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.19999999999999968e-5

   1. Initial program 46.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. flip3-- 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}}, -\tan x\right) \]

      2. associate-/r/ 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}, -\tan x\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{1} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      4. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(\color{blue}{1} + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      5. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \color{blue}{\tan x \cdot \tan \varepsilon}\right)\right), -\tan x\right) \]

      6. +-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}\right), -\tan x\right) \]

      7. pow2 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}\right)\right), -\tan x\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}, -\tan x\right) \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

      2. *-lft-identity 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      3. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      4. distribute-rgt1-in 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      5. fma-def 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)} \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 + \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

   if -5.19999999999999968e-5 < eps < 5.00000000000000024e-5

   1. Initial program 30.1%

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

      1. tan-sum 32.2%

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

      2. div-inv 32.2%

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

   3. Applied egg-rr 32.2%

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

      1. tan-quot 32.2%

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

      2. associate-*r/ 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/l* 32.2%

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

   7. Simplified 32.2%

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

   8. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\left(0.3333333333333333 - \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right)\right)} \]

   10. Simplified 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 - \left(\left(\frac{\sin x}{\frac{\cos x}{\frac{-0.3333333333333333 \cdot \sin x}{\cos x}}} - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-{\sin x}^{3}}{{\cos x}^{3}} - \frac{\sin x}{\cos x}\right)\right)} \]

   if 5.00000000000000024e-5 < eps

   1. Initial program 44.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. div-inv 99.3%

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

      4. sub-neg 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. distribute-lft-neg-in 99.3%

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

      8. add-sqr-sqrt 39.5%

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

      9. sqrt-unprod 68.9%

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

      10. sqr-neg 68.9%

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

      11. sqrt-unprod 29.4%

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

      12. add-sqr-sqrt 48.0%

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

      13. distribute-lft-neg-in 48.0%

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

      14. add-sqr-sqrt 18.6%

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

      15. sqrt-unprod 78.4%

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

      16. sqr-neg 78.4%

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

      17. sqrt-unprod 59.7%

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

   5. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \left(\frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}} - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right), \left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot t_0 + 1}{1 - {t_0}^{3}}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps)))
        (t_1 (- (tan x)))
        (t_2 (+ (tan x) (tan eps))))
   (if (<= eps -6.2e-7)
     (fma
      t_2
      (/ (+ (* (fma (tan x) (tan eps) 1.0) t_0) 1.0) (- 1.0 (pow t_0 3.0)))
      t_1)
     (if (<= eps 4.4e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         eps
         (*
          eps
          (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x))))))
       (fma t_2 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_1)))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = -tan(x);
	double t_2 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -6.2e-7) {
		tmp = fma(t_2, (((fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / (1.0 - pow(t_0, 3.0))), t_1);
	} else if (eps <= 4.4e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (eps * (eps * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x)))));
	} else {
		tmp = fma(t_2, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(-tan(x))
	t_2 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -6.2e-7)
		tmp = fma(t_2, Float64(Float64(Float64(fma(tan(x), tan(eps), 1.0) * t_0) + 1.0) / Float64(1.0 - (t_0 ^ 3.0))), t_1);
	elseif (eps <= 4.4e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(eps * Float64(eps * Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x))))));
	else
		tmp = fma(t_2, Float64(-1.0 / fma(tan(x), tan(eps), -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[Tan[x], $MachinePrecision])}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.2e-7], N[(t$95$2 * N[(N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 4.4e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * 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]), $MachinePrecision], N[(t$95$2 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.1999999999999999e-7

   1. Initial program 46.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. flip3-- 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}}, -\tan x\right) \]

      2. associate-/r/ 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}, -\tan x\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{1} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      4. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(\color{blue}{1} + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right) \]

      5. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \color{blue}{\tan x \cdot \tan \varepsilon}\right)\right), -\tan x\right) \]

      6. +-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}\right), -\tan x\right) \]

      7. pow2 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}\right)\right), -\tan x\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}, -\tan x\right) \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

      2. *-lft-identity 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{1 + \left(\tan x \cdot \tan \varepsilon + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      3. unpow2 99.6%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \left(\tan x \cdot \tan \varepsilon + \color{blue}{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      4. distribute-rgt1-in 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\left(\tan x \cdot \tan \varepsilon + 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

      5. fma-def 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1 + \color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)} \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) \]

   7. Simplified 99.5%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{1 + \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right) \]

   if -6.1999999999999999e-7 < eps < 4.4000000000000002e-7

   1. Initial program 29.8%

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

      1. tan-sum 31.3%

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

      2. div-inv 31.2%

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

   3. Applied egg-rr 31.2%

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

   4. Taylor expanded in eps around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. metadata-eval 99.7%

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

      5. *-lft-identity 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. unpow2 99.7%

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

   6. Simplified 99.7%

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

   if 4.4000000000000002e-7 < eps

   1. Initial program 44.9%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.0%

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

      3. fma-neg 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. frac-2neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. div-inv 99.1%

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

      4. sub-neg 99.1%

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

      5. distribute-neg-in 99.1%

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

      6. metadata-eval 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. add-sqr-sqrt 39.6%

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

      9. sqrt-unprod 69.5%

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

      10. sqr-neg 69.5%

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

      11. sqrt-unprod 29.9%

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

      12. add-sqr-sqrt 48.1%

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

      13. distribute-lft-neg-in 48.1%

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

      14. add-sqr-sqrt 18.2%

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

      15. sqrt-unprod 77.7%

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

      16. sqr-neg 77.7%

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

      17. sqrt-unprod 59.4%

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

   5. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. fma-def 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.05 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.7e-7) (not (<= eps 2.05e-7)))
   (fma
    (+ (tan x) (tan eps))
    (/ -1.0 (fma (tan x) (tan eps) -1.0))
    (- (tan x)))
   (+
    (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
    (*
     eps
     (*
      eps
      (+ (/ (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 <= -3.7e-7) || !(eps <= 2.05e-7)) {
		tmp = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
	} else {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (eps * (eps * ((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 <= -3.7e-7) || !(eps <= 2.05e-7))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	else
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(eps * Float64(eps * 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, -3.7e-7], N[Not[LessEqual[eps, 2.05e-7]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.70000000000000004e-7 or 2.05e-7 < eps

   1. Initial program 45.6%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.2%

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

      3. fma-neg 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. frac-2neg 99.2%

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

      2. metadata-eval 99.2%

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

      3. div-inv 99.2%

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

      4. sub-neg 99.2%

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

      5. distribute-neg-in 99.2%

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

      6. metadata-eval 99.2%

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

      7. distribute-lft-neg-in 99.2%

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

      8. add-sqr-sqrt 43.5%

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

      9. sqrt-unprod 71.1%

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

      10. sqr-neg 71.1%

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

      11. sqrt-unprod 27.5%

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

      12. add-sqr-sqrt 48.0%

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

      13. distribute-lft-neg-in 48.0%

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

      14. add-sqr-sqrt 20.5%

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

      15. sqrt-unprod 76.2%

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

      16. sqr-neg 76.2%

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

      17. sqrt-unprod 55.7%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.3%

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

   7. Simplified 99.3%

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

   if -3.70000000000000004e-7 < eps < 2.05e-7

   1. Initial program 29.8%

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

      1. tan-sum 31.3%

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

      2. div-inv 31.2%

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

   3. Applied egg-rr 31.2%

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

   4. Taylor expanded in eps around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. metadata-eval 99.7%

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

      5. *-lft-identity 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. unpow2 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.05 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.3e-9) (not (<= eps 5.5e-9)))
   (fma
    (+ (tan x) (tan eps))
    (/ -1.0 (fma (tan x) (tan eps) -1.0))
    (- (tan x)))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.3e-9) || !(eps <= 5.5e-9)) {
		tmp = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.3e-9) || !(eps <= 5.5e-9))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.3e-9], N[Not[LessEqual[eps, 5.5e-9]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.29999999999999963e-9 or 5.4999999999999996e-9 < eps

   1. Initial program 45.3%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. fma-neg 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. div-inv 99.0%

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

      4. sub-neg 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft-neg-in 99.0%

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

      8. add-sqr-sqrt 43.2%

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

      9. sqrt-unprod 70.7%

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

      10. sqr-neg 70.7%

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

      11. sqrt-unprod 27.5%

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

      12. add-sqr-sqrt 47.8%

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

      13. distribute-lft-neg-in 47.8%

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

      14. add-sqr-sqrt 20.3%

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

      15. sqrt-unprod 76.1%

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

      16. sqr-neg 76.1%

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

      17. sqrt-unprod 55.8%

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

   5. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

      3. +-commutative 99.0%

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

      4. fma-def 99.1%

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

   7. Simplified 99.1%

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

   if -4.29999999999999963e-9 < eps < 5.4999999999999996e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.6e-9)
     (- (* t_0 (/ -1.0 (fma (tan x) (tan eps) -1.0))) (tan x))
     (if (<= eps 5.4e-9)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (fma t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = (t_0 * (-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x);
	} else if (eps <= 5.4e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(x) * tan(eps)))), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.6e-9)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x));
	elseif (eps <= 5.4e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.6e-9], N[(N[(t$95$0 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.4e-9], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5999999999999998e-9

   1. Initial program 46.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. div-inv 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-neg-in 99.5%

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

      8. add-sqr-sqrt 49.8%

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

      9. sqrt-unprod 73.6%

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

      10. sqr-neg 73.6%

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

      11. sqrt-unprod 23.8%

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

      12. add-sqr-sqrt 47.9%

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

      13. distribute-lft-neg-in 47.9%

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

      14. add-sqr-sqrt 24.2%

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

      15. sqrt-unprod 73.9%

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

      16. sqr-neg 73.9%

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

      17. sqrt-unprod 49.7%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   7. Simplified 99.5%

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

   if -4.5999999999999998e-9 < eps < 5.4000000000000004e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

   if 5.4000000000000004e-9 < eps

   1. Initial program 44.4%

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

      1. tan-sum 98.6%

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

      2. div-inv 98.5%

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

      3. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 8: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.3e-9) (not (<= eps 5.4e-9)))
   (- (* (+ (tan x) (tan eps)) (/ -1.0 (fma (tan x) (tan eps) -1.0))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.3e-9) || !(eps <= 5.4e-9)) {
		tmp = ((tan(x) + tan(eps)) * (-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.3e-9) || !(eps <= 5.4e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) * Float64(-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.3e-9], N[Not[LessEqual[eps, 5.4e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.29999999999999963e-9 or 5.4000000000000004e-9 < eps

   1. Initial program 45.3%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. div-inv 99.0%

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

      4. sub-neg 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft-neg-in 99.0%

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

      8. add-sqr-sqrt 43.2%

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

      9. sqrt-unprod 70.7%

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

      10. sqr-neg 70.7%

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

      11. sqrt-unprod 27.5%

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

      12. add-sqr-sqrt 47.8%

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

      13. distribute-lft-neg-in 47.8%

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

      14. add-sqr-sqrt 20.3%

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

      15. sqrt-unprod 76.1%

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

      16. sqr-neg 76.1%

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

      17. sqrt-unprod 55.8%

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

   5. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

      3. +-commutative 99.0%

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

      4. fma-def 99.1%

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

   7. Simplified 99.0%

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

   if -4.29999999999999963e-9 < eps < 5.4000000000000004e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 9: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.6e-9) (not (<= eps 5.4e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.6e-9) || !(eps <= 5.4e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * (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 <= (-2.6d-9)) .or. (.not. (eps <= 5.4d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((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 <= -2.6e-9) || !(eps <= 5.4e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (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 <= -2.6e-9) or not (eps <= 5.4e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * (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 <= -2.6e-9) || !(eps <= 5.4e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.6e-9) || ~((eps <= 5.4e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.6e-9], N[Not[LessEqual[eps, 5.4e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[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 < -2.6000000000000001e-9 or 5.4000000000000004e-9 < eps

   1. Initial program 45.3%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. fma-neg 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. fma-neg 98.9%

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

      2. associate-*r/ 98.9%

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

      3. *-rgt-identity 98.9%

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

   5. Simplified 98.9%

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

   if -2.6000000000000001e-9 < eps < 5.4000000000000004e-9

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 10: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -8.5e-5)
   (tan eps)
   (if (<= eps 2.7e-5)
     (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5e-5) {
		tmp = tan(eps);
	} else if (eps <= 2.7e-5) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-8.5d-5)) then
        tmp = tan(eps)
    else if (eps <= 2.7d-5) then
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5e-5) {
		tmp = Math.tan(eps);
	} else if (eps <= 2.7e-5) {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -8.5e-5:
		tmp = math.tan(eps)
	elif eps <= 2.7e-5:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -8.5e-5)
		tmp = tan(eps);
	elseif (eps <= 2.7e-5)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -8.5e-5)
		tmp = tan(eps);
	elseif (eps <= 2.7e-5)
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -8.5e-5], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 2.7e-5], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.500000000000001e-5 or 2.6999999999999999e-5 < eps

   1. Initial program 45.5%

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

      1. add-log-exp 45.3%

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

   3. Applied egg-rr 45.3%

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

   4. Taylor expanded in x around 0 47.5%

      \[\leadsto \log \color{blue}{\left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 47.5%

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

      2. log-prod 47.5%

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

      3. metadata-eval 47.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

      4. add-log-exp 47.7%

         \[\leadsto 0 + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

      5. tan-quot 47.9%

         \[\leadsto 0 + \color{blue}{\tan \varepsilon} \]

   6. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{0 + \tan \varepsilon} \]
   7. Step-by-step derivation

      1. +-lft-identity 47.9%

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

   8. Simplified 47.9%

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

   if -8.500000000000001e-5 < eps < 2.6999999999999999e-5

   1. Initial program 30.1%

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

      1. tan-sum 32.2%

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

      2. div-inv 32.2%

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

      3. fma-neg 32.2%

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

   3. Applied egg-rr 32.2%

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

   4. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.5e-6)
   (tan eps)
   (if (<= eps 2.9e-5)
     (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-6) {
		tmp = tan(eps);
	} else if (eps <= 2.9e-5) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-3.5d-6)) then
        tmp = tan(eps)
    else if (eps <= 2.9d-5) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 2.9e-5) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.5e-6:
		tmp = math.tan(eps)
	elif eps <= 2.9e-5:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.5e-6)
		tmp = tan(eps);
	elseif (eps <= 2.9e-5)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.5e-6)
		tmp = tan(eps);
	elseif (eps <= 2.9e-5)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.5e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 2.9e-5], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.49999999999999995e-6 or 2.9e-5 < eps

   1. Initial program 45.5%

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

      1. add-log-exp 45.3%

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

   3. Applied egg-rr 45.3%

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

   4. Taylor expanded in x around 0 47.5%

      \[\leadsto \log \color{blue}{\left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 47.5%

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

      2. log-prod 47.5%

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

      3. metadata-eval 47.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

      4. add-log-exp 47.7%

         \[\leadsto 0 + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

      5. tan-quot 47.9%

         \[\leadsto 0 + \color{blue}{\tan \varepsilon} \]

   6. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{0 + \tan \varepsilon} \]
   7. Step-by-step derivation

      1. +-lft-identity 47.9%

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

   8. Simplified 47.9%

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

   if -3.49999999999999995e-6 < eps < 2.9e-5

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. metadata-eval 98.4%

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

      3. *-lft-identity 98.4%

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

      4. distribute-lft-in 98.5%

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

      5. *-rgt-identity 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12: 57.5% 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 37.8%

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

   1. add-log-exp 26.1%

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

3. Applied egg-rr 26.1%

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

4. Taylor expanded in x around 0 27.6%

   \[\leadsto \log \color{blue}{\left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \]
5. Step-by-step derivation

   1. *-un-lft-identity 27.6%

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

   2. log-prod 27.6%

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

   3. metadata-eval 27.6%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

   4. add-log-exp 52.8%

      \[\leadsto 0 + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   5. tan-quot 52.9%

      \[\leadsto 0 + \color{blue}{\tan \varepsilon} \]

6. Applied egg-rr 52.9%

   \[\leadsto \color{blue}{0 + \tan \varepsilon} \]
7. Step-by-step derivation

   1. +-lft-identity 52.9%

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

8. Simplified 52.9%

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

9. Final simplification 52.9%

   \[\leadsto \tan \varepsilon \]

Alternative 13: 30.6% 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 37.8%

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

   1. add-log-exp 26.1%

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

3. Applied egg-rr 26.1%

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

4. Taylor expanded in x around 0 27.6%

   \[\leadsto \log \color{blue}{\left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)} \]

5. Taylor expanded in eps around 0 30.7%

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

6. Final simplification 30.7%

   \[\leadsto \varepsilon \]

Developer target: 76.0% 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 2023182 
(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)))