2tan (problem 3.3.2)

Percentage Accurate: 41.5% → 99.5%

Time: 27.7s

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: 41.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1 \cdot \cos x + \sin x \cdot \left(t_0 + -1\right)}{\cos x \cdot \left(1 - t_0\right)}\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}\right)} + \left({\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \left(\sin x \cdot \frac{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}{\cos x} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -5.2e-5)
     (/ (+ (* t_1 (cos x)) (* (sin x) (+ t_0 -1.0))) (* (cos x) (- 1.0 t_0)))
     (if (<= eps 5.4e-5)
       (+
        (/
         (sin eps)
         (* (cos eps) (- 1.0 (/ (/ (* (sin x) (sin eps)) (cos eps)) (cos x)))))
        (+
         (* (pow (sin x) 2.0) (/ eps (pow (cos x) 2.0)))
         (-
          (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
          (*
           (pow eps 3.0)
           (-
            (* (sin x) (/ (* (/ (sin x) (cos x)) -0.3333333333333333) (cos x)))
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))))
       (- (/ t_1 (- (fma (tan x) (tan eps) -1.0))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -5.2e-5) {
		tmp = ((t_1 * cos(x)) + (sin(x) * (t_0 + -1.0))) / (cos(x) * (1.0 - t_0));
	} else if (eps <= 5.4e-5) {
		tmp = (sin(eps) / (cos(eps) * (1.0 - (((sin(x) * sin(eps)) / cos(eps)) / cos(x))))) + ((pow(sin(x), 2.0) * (eps / pow(cos(x), 2.0))) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) - (pow(eps, 3.0) * ((sin(x) * (((sin(x) / cos(x)) * -0.3333333333333333) / cos(x))) - (pow(sin(x), 4.0) / pow(cos(x), 4.0))))));
	} else {
		tmp = (t_1 / -fma(tan(x), tan(eps), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -5.2e-5)
		tmp = Float64(Float64(Float64(t_1 * cos(x)) + Float64(sin(x) * Float64(t_0 + -1.0))) / Float64(cos(x) * Float64(1.0 - t_0)));
	elseif (eps <= 5.4e-5)
		tmp = Float64(Float64(sin(eps) / Float64(cos(eps) * Float64(1.0 - Float64(Float64(Float64(sin(x) * sin(eps)) / cos(eps)) / cos(x))))) + Float64(Float64((sin(x) ^ 2.0) * Float64(eps / (cos(x) ^ 2.0))) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) - Float64((eps ^ 3.0) * Float64(Float64(sin(x) * Float64(Float64(Float64(sin(x) / cos(x)) * -0.3333333333333333) / cos(x))) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))))));
	else
		tmp = Float64(Float64(t_1 / Float64(-fma(tan(x), tan(eps), -1.0))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.2e-5], N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.4e-5], N[(N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(eps / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / (-N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.19999999999999968e-5

   1. Initial program 51.6%

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

      1. tan-sum 99.4%

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

      2. tan-quot 99.2%

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

      3. frac-sub 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -5.19999999999999968e-5 < eps < 5.3999999999999998e-5

   1. Initial program 26.1%

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

      1. tan-sum 27.8%

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

      2. div-inv 27.8%

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

      3. fma-neg 27.8%

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

   3. Applied egg-rr 27.8%

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

   4. Taylor expanded in x around inf 27.7%

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

      1. associate--l+ 54.4%

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

      2. associate-/r* 54.4%

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

      3. *-commutative 54.4%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in eps around 0 99.7%

      \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon}}{\cos x}\right)} + \color{blue}{\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. *-lft-identity 99.7%

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

      3. times-frac 99.7%

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

      4. /-rgt-identity 99.7%

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

      5. mul-1-neg 99.7%

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

   9. Simplified 99.7%

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

   if 5.3999999999999998e-5 < eps

   1. Initial program 47.2%

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

      1. tan-sum 99.4%

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

      2. clear-num 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 85.7%

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

      2. expm1-udef 85.8%

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

      3. log1p-udef 85.8%

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

      4. add-exp-log 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. associate--l+ 99.3%

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

      2. fma-neg 99.3%

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

      3. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-/r/ 99.4%

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

      3. associate--r+ 99.5%

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

      4. metadata-eval 99.5%

         \[\leadsto \frac{1}{\color{blue}{0} - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) + \left(-\tan x\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{1}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) + \left(-\tan x\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.5%

          \[\leadsto \color{blue}{\frac{1}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x} \]

       2. associate-*l/ 99.5%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\tan x + \tan \varepsilon\right)}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]

       3. *-lft-identity 99.5%

          \[\leadsto \frac{\color{blue}{\tan x + \tan \varepsilon}}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x \]

       4. sub0-neg 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x + \sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}\right)} + \left({\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \left(\sin x \cdot \frac{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}{\cos x} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1 \cdot \cos x + \sin x \cdot \left(t_0 + -1\right)}{\cos x \cdot \left(1 - t_0\right)}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -9.2e-7)
     (/ (+ (* t_1 (cos x)) (* (sin x) (+ t_0 -1.0))) (* (cos x) (- 1.0 t_0)))
     (if (<= eps 4.2e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- (/ t_1 (- (fma (tan x) (tan eps) -1.0))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -9.2e-7) {
		tmp = ((t_1 * cos(x)) + (sin(x) * (t_0 + -1.0))) / (cos(x) * (1.0 - t_0));
	} else if (eps <= 4.2e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_1 / -fma(tan(x), tan(eps), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -9.2e-7)
		tmp = Float64(Float64(Float64(t_1 * cos(x)) + Float64(sin(x) * Float64(t_0 + -1.0))) / Float64(cos(x) * Float64(1.0 - t_0)));
	elseif (eps <= 4.2e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_1 / Float64(-fma(tan(x), tan(eps), -1.0))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -9.2e-7], N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.2e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / (-N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -9.1999999999999998e-7

   1. Initial program 51.6%

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

      1. tan-sum 99.4%

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

      2. tan-quot 99.2%

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

      3. frac-sub 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -9.1999999999999998e-7 < eps < 4.2e-7

   1. Initial program 26.1%

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

      1. tan-sum 27.8%

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

      2. clear-num 27.5%

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

   3. Applied egg-rr 27.5%

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

   4. Taylor expanded in eps around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. metadata-eval 99.4%

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

      5. *-lft-identity 99.4%

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

      6. distribute-lft-in 99.5%

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

      7. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 4.2e-7 < eps

   1. Initial program 47.2%

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

      1. tan-sum 99.4%

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

      2. clear-num 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 85.7%

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

      2. expm1-udef 85.8%

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

      3. log1p-udef 85.8%

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

      4. add-exp-log 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. associate--l+ 99.3%

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

      2. fma-neg 99.3%

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

      3. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-/r/ 99.4%

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

      3. associate--r+ 99.5%

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

      4. metadata-eval 99.5%

         \[\leadsto \frac{1}{\color{blue}{0} - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) + \left(-\tan x\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{1}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) + \left(-\tan x\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.5%

          \[\leadsto \color{blue}{\frac{1}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x} \]

       2. associate-*l/ 99.5%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\tan x + \tan \varepsilon\right)}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]

       3. *-lft-identity 99.5%

          \[\leadsto \frac{\color{blue}{\tan x + \tan \varepsilon}}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x \]

       4. sub0-neg 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x + \sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7e-9) (not (<= eps 3e-9)))
   (fma
    (+ (tan x) (tan eps))
    (/ 1.0 (- 1.0 (* (tan x) (tan eps))))
    (- (tan x)))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-9) || !(eps <= 3e-9)) {
		tmp = fma((tan(x) + tan(eps)), (1.0 / (1.0 - (tan(x) * tan(eps)))), -tan(x));
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.7e-9) || !(eps <= 3e-9))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), Float64(-tan(x)));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.7e-9], N[Not[LessEqual[eps, 3e-9]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.6999999999999999e-9 or 2.99999999999999998e-9 < eps

   1. Initial program 48.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 98.6%

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

      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%

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

   if -1.6999999999999999e-9 < eps < 2.99999999999999998e-9

   1. Initial program 26.6%

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

      1. tan-sum 26.9%

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

      2. div-inv 26.9%

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

      3. fma-neg 26.9%

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

   3. Applied egg-rr 26.9%

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

   4. Taylor expanded in eps around 0 99.3%

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

      1. distribute-rgt-in 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. unpow2 99.4%

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

      4. unpow2 99.4%

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

      5. frac-times 99.4%

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

      6. tan-quot 99.5%

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

      7. tan-quot 99.4%

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

      8. pow2 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. fma-def 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.8e-9)
     (- (* t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps))))) (tan x))
     (if (<= eps 3.6e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 (- (fma (tan x) (tan eps) -1.0))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.8e-9) {
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else if (eps <= 3.6e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / -fma(tan(x), tan(eps), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.8e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps))))) - tan(x));
	elseif (eps <= 3.6e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(x), tan(eps), -1.0))) - 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, -4.8e-9], N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.6e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(t$95$0 / (-N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.8e-9

   1. Initial program 50.8%

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

      1. tan-sum 53.9%

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

      2. div-inv 53.9%

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

   3. Applied egg-rr 99.0%

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

   if -4.8e-9 < eps < 3.6e-9

   1. Initial program 26.5%

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

      1. tan-sum 27.1%

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

      2. div-inv 27.1%

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

      3. fma-neg 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in eps around 0 99.0%

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

      1. distribute-rgt-in 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. unpow2 99.2%

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

      4. unpow2 99.2%

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

      5. frac-times 99.1%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.2%

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

      8. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-def 99.2%

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

   8. Simplified 99.2%

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

   if 3.6e-9 < eps

   1. Initial program 46.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. clear-num 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. expm1-log1p-u 85.4%

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

      2. expm1-udef 85.5%

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

      3. log1p-udef 85.5%

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

      4. add-exp-log 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. associate--l+ 98.8%

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

      2. fma-neg 98.8%

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

      3. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

      1. sub-neg 98.8%

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

      2. associate-/r/ 98.9%

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

      3. associate--r+ 99.0%

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

      4. metadata-eval 99.0%

         \[\leadsto \frac{1}{\color{blue}{0} - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) + \left(-\tan x\right) \]

   9. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{1}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) + \left(-\tan x\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.0%

          \[\leadsto \color{blue}{\frac{1}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x} \]

       2. associate-*l/ 98.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(\tan x + \tan \varepsilon\right)}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]

       3. *-lft-identity 98.9%

          \[\leadsto \frac{\color{blue}{\tan x + \tan \varepsilon}}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x \]

       4. sub0-neg 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.6e-9) (not (<= eps 4.2e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.6e-9) || !(eps <= 4.2e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.6e-9) || !(eps <= 4.2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.6e-9], N[Not[LessEqual[eps, 4.2e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.6000000000000001e-9 or 4.20000000000000039e-9 < eps

   1. Initial program 48.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 < 4.20000000000000039e-9

   1. Initial program 26.5%

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

      1. tan-sum 27.1%

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

      2. div-inv 27.1%

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

      3. fma-neg 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in eps around 0 99.0%

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

      1. distribute-rgt-in 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. unpow2 99.2%

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

      4. unpow2 99.2%

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

      5. frac-times 99.1%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.2%

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

      8. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-def 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -4.3e-9)
     (- (* t_0 (/ 1.0 t_1)) (tan x))
     (if (<= eps 3.7e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -4.3e-9) {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	} else if (eps <= 3.7e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / t_1) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -4.3e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	elseif (eps <= 3.7e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.3e-9], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.7e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.29999999999999963e-9

   1. Initial program 50.8%

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

      1. tan-sum 53.9%

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

      2. div-inv 53.9%

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

   3. Applied egg-rr 99.0%

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

   if -4.29999999999999963e-9 < eps < 3.7e-9

   1. Initial program 26.5%

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

      1. tan-sum 27.1%

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

      2. div-inv 27.1%

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

      3. fma-neg 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in eps around 0 99.0%

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

      1. distribute-rgt-in 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. unpow2 99.2%

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

      4. unpow2 99.2%

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

      5. frac-times 99.1%

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

      6. tan-quot 99.3%

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

      7. tan-quot 99.2%

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

      8. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-def 99.2%

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

   8. Simplified 99.2%

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

   if 3.7e-9 < eps

   1. Initial program 46.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 98.8%

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

      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 98.8%

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

      2. associate-*r/ 98.9%

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

      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%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 7: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000112:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00155)
   (tan eps)
   (if (<= eps 0.000112)
     (fma eps (pow (tan x) 2.0) eps)
     (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = tan(eps);
	} else if (eps <= 0.000112) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 0.000112)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00155], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.000112], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00154999999999999995

   1. Initial program 52.4%

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

   2. Taylor expanded in x around 0 55.5%

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

      1. tan-quot 55.8%

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

      2. expm1-log1p-u 40.7%

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

      3. expm1-udef 40.4%

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

   4. Applied egg-rr 40.4%

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

      1. expm1-def 40.7%

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

      2. expm1-log1p 55.8%

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

   6. Simplified 55.8%

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

   if -0.00154999999999999995 < eps < 1.11999999999999998e-4

   1. Initial program 26.0%

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

      1. tan-sum 28.2%

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

      2. div-inv 28.2%

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

      3. fma-neg 28.3%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. distribute-rgt-in 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. unpow2 98.0%

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

      4. unpow2 98.0%

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

      5. frac-times 98.0%

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

      6. tan-quot 98.1%

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

      7. tan-quot 98.0%

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

      8. pow2 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. *-commutative 98.0%

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

      3. fma-def 98.0%

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

   8. Simplified 98.0%

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

   if 1.11999999999999998e-4 < eps

   1. Initial program 47.2%

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

      1. add-cbrt-cube 47.0%

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

      2. pow1/3 22.5%

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

      3. pow3 22.3%

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

      4. pow-to-exp 22.4%

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

      5. pow-exp 22.3%

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

   3. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{e^{\left(\log \tan \left(x + \varepsilon\right) \cdot 3\right) \cdot 0.3333333333333333}} - \tan x \]
   4. Step-by-step derivation

      1. associate-*l* 22.3%

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

      2. metadata-eval 22.3%

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

      3. pow-to-exp 47.2%

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

      4. pow1 47.2%

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

      5. add-sqr-sqrt 22.1%

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

      6. sqrt-unprod 27.2%

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

      7. pow2 27.2%

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

   5. Applied egg-rr 27.2%

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

      1. unpow2 27.2%

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

      2. rem-sqrt-square 27.2%

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

      3. +-commutative 27.2%

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

   7. Simplified 27.2%

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

   8. Taylor expanded in x around 0 27.9%

      \[\leadsto \color{blue}{\left|\tan \left(\varepsilon + x\right)\right|} \]
   9. Step-by-step derivation

      1. +-commutative 27.9%

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

      2. unpow1 27.9%

         \[\leadsto \left|\color{blue}{{\tan \left(x + \varepsilon\right)}^{1}}\right| \]

      3. sqr-pow 21.7%

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

      4. fabs-sqr 21.7%

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

      5. sqr-pow 46.9%

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

      6. unpow1 46.9%

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

   10. Simplified 46.9%

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

       1. tan-sum 52.3%

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

       2. div-inv 52.3%

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

   12. Applied egg-rr 52.3%

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

       1. associate-*r/ 52.3%

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

       2. *-rgt-identity 52.3%

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

   14. Simplified 52.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000112:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\\ \end{array} \]

Alternative 8: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00155)
   (tan eps)
   (if (<= eps 6.5e-5)
     (fma eps (pow (tan x) 2.0) eps)
     (- (/ 1.0 (/ (cos eps) (sin eps))) (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = tan(eps);
	} else if (eps <= 6.5e-5) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 6.5e-5)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(1.0 / Float64(cos(eps) / sin(eps))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00155], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 6.5e-5], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(1.0 / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00154999999999999995

   1. Initial program 52.4%

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

   2. Taylor expanded in x around 0 55.5%

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

      1. tan-quot 55.8%

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

      2. expm1-log1p-u 40.7%

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

      3. expm1-udef 40.4%

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

   4. Applied egg-rr 40.4%

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

      1. expm1-def 40.7%

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

      2. expm1-log1p 55.8%

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

   6. Simplified 55.8%

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

   if -0.00154999999999999995 < eps < 6.49999999999999943e-5

   1. Initial program 26.0%

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

      1. tan-sum 28.2%

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

      2. div-inv 28.2%

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

      3. fma-neg 28.3%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. distribute-rgt-in 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. unpow2 98.0%

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

      4. unpow2 98.0%

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

      5. frac-times 98.0%

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

      6. tan-quot 98.1%

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

      7. tan-quot 98.0%

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

      8. pow2 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. *-commutative 98.0%

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

      3. fma-def 98.0%

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

   8. Simplified 98.0%

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

   if 6.49999999999999943e-5 < eps

   1. Initial program 47.2%

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

      1. tan-sum 99.4%

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

      2. clear-num 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x around 0 50.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \end{array} \]

Alternative 9: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00155)
   (tan eps)
   (if (<= eps 2.9e-6) (fma eps (pow (tan x) 2.0) eps) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = tan(eps);
	} else if (eps <= 2.9e-6) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 2.9e-6)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = tan(eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00155], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 2.9e-6], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00154999999999999995 or 2.9000000000000002e-6 < eps

   1. Initial program 49.8%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. tan-quot 52.9%

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

      2. expm1-log1p-u 39.6%

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

      3. expm1-udef 39.4%

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

   4. Applied egg-rr 39.4%

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

      1. expm1-def 39.6%

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

      2. expm1-log1p 52.9%

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

   6. Simplified 52.9%

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

   if -0.00154999999999999995 < eps < 2.9000000000000002e-6

   1. Initial program 26.0%

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

      1. tan-sum 28.2%

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

      2. div-inv 28.2%

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

      3. fma-neg 28.3%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. distribute-rgt-in 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. unpow2 98.0%

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

      4. unpow2 98.0%

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

      5. frac-times 98.0%

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

      6. tan-quot 98.1%

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

      7. tan-quot 98.0%

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

      8. pow2 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. *-commutative 98.0%

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

      3. fma-def 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 10: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00155)
   (tan eps)
   (if (<= eps 1.05e-5) (* eps (+ 1.0 (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = tan(eps);
	} else if (eps <= 1.05e-5) {
		tmp = eps * (1.0 + pow(tan(x), 2.0));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.00155d0)) then
        tmp = tan(eps)
    else if (eps <= 1.05d-5) then
        tmp = eps * (1.0d0 + (tan(x) ** 2.0d0))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = Math.tan(eps);
	} else if (eps <= 1.05e-5) {
		tmp = eps * (1.0 + Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00155:
		tmp = math.tan(eps)
	elif eps <= 1.05e-5:
		tmp = eps * (1.0 + math.pow(math.tan(x), 2.0))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 1.05e-5)
		tmp = Float64(eps * Float64(1.0 + (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 1.05e-5)
		tmp = eps * (1.0 + (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00155], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1.05e-5], N[(eps * N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00154999999999999995 or 1.04999999999999994e-5 < eps

   1. Initial program 49.8%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. tan-quot 52.9%

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

      2. expm1-log1p-u 39.6%

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

      3. expm1-udef 39.4%

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

   4. Applied egg-rr 39.4%

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

      1. expm1-def 39.6%

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

      2. expm1-log1p 52.9%

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

   6. Simplified 52.9%

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

   if -0.00154999999999999995 < eps < 1.04999999999999994e-5

   1. Initial program 26.0%

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

      1. tan-sum 28.2%

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

      2. div-inv 28.2%

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

      3. fma-neg 28.3%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. unpow2 97.9%

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

      2. unpow2 97.9%

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

      3. frac-times 97.9%

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

      4. tan-quot 97.9%

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

      5. tan-quot 97.9%

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

      6. *-un-lft-identity 97.9%

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

      7. pow2 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00155)
   (tan eps)
   (if (<= eps 3.35e-6) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = tan(eps);
	} else if (eps <= 3.35e-6) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.00155d0)) then
        tmp = tan(eps)
    else if (eps <= 3.35d-6) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00155) {
		tmp = Math.tan(eps);
	} else if (eps <= 3.35e-6) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00155:
		tmp = math.tan(eps)
	elif eps <= 3.35e-6:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 3.35e-6)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00155)
		tmp = tan(eps);
	elseif (eps <= 3.35e-6)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00155], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.35e-6], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00154999999999999995 or 3.35e-6 < eps

   1. Initial program 49.8%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. tan-quot 52.9%

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

      2. expm1-log1p-u 39.6%

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

      3. expm1-udef 39.4%

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

   4. Applied egg-rr 39.4%

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

      1. expm1-def 39.6%

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

      2. expm1-log1p 52.9%

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

   6. Simplified 52.9%

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

   if -0.00154999999999999995 < eps < 3.35e-6

   1. Initial program 26.0%

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

      1. tan-sum 28.2%

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

      2. div-inv 28.2%

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

      3. fma-neg 28.3%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. distribute-rgt-in 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. unpow2 98.0%

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

      4. unpow2 98.0%

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

      5. frac-times 98.0%

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

      6. tan-quot 98.1%

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

      7. tan-quot 98.0%

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

      8. pow2 98.0%

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

   6. Applied egg-rr 98.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12: 57.7% 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 37.0%

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

2. Taylor expanded in x around 0 52.7%

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

   1. tan-quot 52.8%

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

   2. expm1-log1p-u 46.7%

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

   3. expm1-udef 21.7%

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

4. Applied egg-rr 21.7%

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

   1. expm1-def 46.7%

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

   2. expm1-log1p 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

   \[\leadsto \tan \varepsilon \]

Alternative 13: 31.0% 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 37.0%

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

2. Taylor expanded in x around 0 52.7%

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

3. Taylor expanded in eps around 0 29.9%

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

4. Final simplification 29.9%

   \[\leadsto \varepsilon \]

Developer target: 76.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(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)))