2tan (problem 3.3.2)

Percentage Accurate: 42.5% → 99.5%

Time: 21.9s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.5% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.1e-7)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 3.3e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))))
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.1e-7) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 3.3e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))));
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.1e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 3.3e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x)))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.1e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.3e-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[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.0999999999999999e-7

   1. Initial program 52.7%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. fma-neg 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

   if -4.0999999999999999e-7 < eps < 3.3000000000000002e-7

   1. Initial program 35.2%

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

      1. tan-sum 36.0%

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

      2. div-inv 36.0%

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

      3. fma-neg 36.0%

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

   3. Applied egg-rr 36.0%

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

      1. fma-neg 36.0%

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

      2. associate-*r/ 36.0%

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

      3. *-rgt-identity 36.0%

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

   5. Simplified 36.0%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

   8. Simplified 99.6%

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

   if 3.3000000000000002e-7 < eps

   1. Initial program 58.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. div-inv 99.4%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 47.6%

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

      8. sqrt-unprod 81.2%

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

      9. sqr-neg 81.2%

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

      10. sqrt-unprod 33.5%

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

      11. add-sqr-sqrt 59.8%

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

      12. distribute-lft-neg-in 59.8%

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

      13. add-sqr-sqrt 26.2%

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

      14. sqrt-unprod 78.0%

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

      15. sqr-neg 78.0%

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

      16. sqrt-unprod 51.7%

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

      17. add-sqr-sqrt 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -12500:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -12500.0)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 9e-9)
       (+
        (/
         (sin eps)
         (* (cos eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x))))))
        (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0))))
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -12500.0) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 9e-9) {
		tmp = (sin(eps) / (cos(eps) * (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x)))))) + (eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0)));
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -12500.0)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 9e-9)
		tmp = Float64(Float64(sin(eps) / Float64(cos(eps) * Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x)))))) + Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -12500.0], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 9e-9], N[(N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -12500

   1. Initial program 52.0%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. fma-neg 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

   if -12500 < eps < 8.99999999999999953e-9

   1. Initial program 35.7%

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

      1. tan-sum 36.5%

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

      2. div-inv 36.5%

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

      3. fma-neg 36.5%

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

   3. Applied egg-rr 36.5%

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

      1. fma-neg 36.5%

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

      2. associate-*r/ 36.5%

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

      3. *-rgt-identity 36.5%

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

   5. Simplified 36.5%

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

   6. Taylor expanded in x around inf 36.5%

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

      1. associate--l+ 58.6%

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

      2. times-frac 58.6%

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

      3. associate-/r* 58.6%

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

      4. times-frac 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in eps around 0 99.4%

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

       1. associate-/l* 99.4%

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

   11. Simplified 99.4%

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

   if 8.99999999999999953e-9 < eps

   1. Initial program 58.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. div-inv 99.4%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 47.6%

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

      8. sqrt-unprod 81.2%

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

      9. sqr-neg 81.2%

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

      10. sqrt-unprod 33.5%

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

      11. add-sqr-sqrt 59.8%

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

      12. distribute-lft-neg-in 59.8%

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

      13. add-sqr-sqrt 26.2%

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

      14. sqrt-unprod 78.0%

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

      15. sqr-neg 78.0%

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

      16. sqrt-unprod 51.7%

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

      17. add-sqr-sqrt 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -12500:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -12500:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -12500.0)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 5.2e-9)
       (+
        (/
         (sin eps)
         (* (cos eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x))))))
        (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0)))
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -12500.0) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 5.2e-9) {
		tmp = (sin(eps) / (cos(eps) * (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x)))))) + ((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0));
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -12500.0)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 5.2e-9)
		tmp = Float64(Float64(sin(eps) / Float64(cos(eps) * Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x)))))) + Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -12500.0], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-9], N[(N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -12500

   1. Initial program 52.0%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. fma-neg 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

   if -12500 < eps < 5.2000000000000002e-9

   1. Initial program 35.7%

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

      1. tan-sum 36.5%

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

      2. div-inv 36.5%

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

      3. fma-neg 36.5%

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

   3. Applied egg-rr 36.5%

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

      1. fma-neg 36.5%

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

      2. associate-*r/ 36.5%

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

      3. *-rgt-identity 36.5%

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

   5. Simplified 36.5%

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

   6. Taylor expanded in x around inf 36.5%

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

      1. associate--l+ 58.6%

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

      2. times-frac 58.6%

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

      3. associate-/r* 58.6%

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

      4. times-frac 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in eps around 0 99.4%

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

   if 5.2000000000000002e-9 < eps

   1. Initial program 58.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. div-inv 99.4%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 47.6%

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

      8. sqrt-unprod 81.2%

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

      9. sqr-neg 81.2%

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

      10. sqrt-unprod 33.5%

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

      11. add-sqr-sqrt 59.8%

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

      12. distribute-lft-neg-in 59.8%

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

      13. add-sqr-sqrt 26.2%

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

      14. sqrt-unprod 78.0%

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

      15. sqr-neg 78.0%

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

      16. sqrt-unprod 51.7%

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

      17. add-sqr-sqrt 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -12500:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -7e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 4.2e-9)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -7e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 4.2e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -7e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 4.2e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.2e-9], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.9999999999999998e-9

   1. Initial program 52.7%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

      3. fma-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. fma-neg 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

   if -6.9999999999999998e-9 < eps < 4.20000000000000039e-9

   1. Initial program 35.2%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

   if 4.20000000000000039e-9 < eps

   1. Initial program 58.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. div-inv 99.4%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 47.6%

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

      8. sqrt-unprod 81.2%

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

      9. sqr-neg 81.2%

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

      10. sqrt-unprod 33.5%

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

      11. add-sqr-sqrt 59.8%

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

      12. distribute-lft-neg-in 59.8%

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

      13. add-sqr-sqrt 26.2%

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

      14. sqrt-unprod 78.0%

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

      15. sqr-neg 78.0%

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

      16. sqrt-unprod 51.7%

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

      17. add-sqr-sqrt 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.7e-9) (not (<= eps 3e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-9) || !(eps <= 3e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.7d-9)) .or. (.not. (eps <= 3d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-9) || !(eps <= 3e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.7e-9) or not (eps <= 3e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.7e-9) || !(eps <= 3e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.7e-9) || ~((eps <= 3e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.7e-9], N[Not[LessEqual[eps, 3e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.7e-9 or 2.99999999999999998e-9 < eps

   1. Initial program 55.6%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.6%

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

      3. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

   if -3.7e-9 < eps < 2.99999999999999998e-9

   1. Initial program 35.2%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.9e-6)
   (tan eps)
   (if (<= eps 3e-6)
     (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.9e-6) {
		tmp = tan(eps);
	} else if (eps <= 3e-6) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.9d-6)) then
        tmp = tan(eps)
    else if (eps <= 3d-6) then
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.9e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 3e-6) {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.9e-6:
		tmp = math.tan(eps)
	elif eps <= 3e-6:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.9e-6)
		tmp = tan(eps);
	elseif (eps <= 3e-6)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.9e-6)
		tmp = tan(eps);
	elseif (eps <= 3e-6)
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.9e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3e-6], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.9e-6 or 3.0000000000000001e-6 < eps

   1. Initial program 55.6%

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

   2. Taylor expanded in x around 0 58.0%

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

      1. tan-quot 58.2%

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

      2. expm1-log1p-u 42.0%

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

      3. expm1-udef 41.6%

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

   4. Applied egg-rr 41.6%

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

      1. expm1-def 42.0%

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

      2. expm1-log1p 58.2%

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

   6. Simplified 58.2%

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

   if -1.9e-6 < eps < 3.0000000000000001e-6

   1. Initial program 35.2%

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

      1. tan-sum 36.0%

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

      2. div-inv 36.0%

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

      3. fma-neg 36.0%

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

   3. Applied egg-rr 36.0%

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

   4. Taylor expanded in eps around 0 99.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.9 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.9e-6)
   (tan eps)
   (if (<= eps 1e-5)
     (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5.9e-6) {
		tmp = tan(eps);
	} else if (eps <= 1e-5) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-5.9d-6)) then
        tmp = tan(eps)
    else if (eps <= 1d-5) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -5.9e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 1e-5) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -5.9e-6:
		tmp = math.tan(eps)
	elif eps <= 1e-5:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5.9e-6)
		tmp = tan(eps);
	elseif (eps <= 1e-5)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -5.9e-6)
		tmp = tan(eps);
	elseif (eps <= 1e-5)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5.9e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1e-5], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.90000000000000026e-6 or 1.00000000000000008e-5 < eps

   1. Initial program 55.6%

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

   2. Taylor expanded in x around 0 58.0%

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

      1. tan-quot 58.2%

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

      2. expm1-log1p-u 42.0%

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

      3. expm1-udef 41.6%

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

   4. Applied egg-rr 41.6%

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

      1. expm1-def 42.0%

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

      2. expm1-log1p 58.2%

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

   6. Simplified 58.2%

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

   if -5.90000000000000026e-6 < eps < 1.00000000000000008e-5

   1. Initial program 35.2%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. *-rgt-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.9 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8: 57.9% 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 46.1%

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

2. Taylor expanded in x around 0 58.0%

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

   1. tan-quot 58.0%

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

   2. expm1-log1p-u 49.4%

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

   3. expm1-udef 25.4%

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

4. Applied egg-rr 25.4%

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

   1. expm1-def 49.4%

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

   2. expm1-log1p 58.0%

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

6. Simplified 58.0%

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

7. Final simplification 58.0%

   \[\leadsto \tan \varepsilon \]

Alternative 9: 30.7% 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 46.1%

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

2. Taylor expanded in x around 0 58.0%

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

3. Taylor expanded in eps around 0 28.7%

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

4. Final simplification 28.7%

   \[\leadsto \varepsilon \]

Developer target: 76.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023280 
(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)))