2tan (problem 3.3.2)

Percentage Accurate: 42.5% → 99.5%

Time: 15.3s

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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_3 := \frac{\sin x \cdot -0.3333333333333333}{\cos x}\\ t_4 := \frac{\sin x}{\frac{\cos x}{t_3}}\\ t_5 := \tan x + \tan \varepsilon\\ t_6 := {\cos x}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.0305:\\ \;\;\;\;\mathsf{fma}\left(t_5, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_2}{1 - t_2 \cdot \frac{\sin x}{\cos x}} + \left(\mathsf{fma}\left(-{\varepsilon}^{4}, \frac{t_1}{\frac{t_6}{t_3}} + \frac{\sin x}{\frac{\cos x}{t_4 - t_0}}, \frac{\varepsilon}{\frac{t_6}{t_1}}\right) + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \left(t_0 - t_4\right) \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_5}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (/ (sin eps) (cos eps)))
        (t_3 (/ (* (sin x) -0.3333333333333333) (cos x)))
        (t_4 (/ (sin x) (/ (cos x) t_3)))
        (t_5 (+ (tan x) (tan eps)))
        (t_6 (pow (cos x) 2.0)))
   (if (<= eps -0.0305)
     (fma t_5 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) (- (tan x)))
     (if (<= eps 2.45e-6)
       (+
        (/ t_2 (- 1.0 (* t_2 (/ (sin x) (cos x)))))
        (+
         (fma
          (- (pow eps 4.0))
          (+ (/ t_1 (/ t_6 t_3)) (/ (sin x) (/ (cos x) (- t_4 t_0))))
          (/ eps (/ t_6 t_1)))
         (+
          (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
          (* (- t_0 t_4) (pow eps 3.0)))))
       (-
        (/ t_5 (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 4.0) / pow(cos(x), 4.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sin(eps) / cos(eps);
	double t_3 = (sin(x) * -0.3333333333333333) / cos(x);
	double t_4 = sin(x) / (cos(x) / t_3);
	double t_5 = tan(x) + tan(eps);
	double t_6 = pow(cos(x), 2.0);
	double tmp;
	if (eps <= -0.0305) {
		tmp = fma(t_5, (1.0 / (1.0 - (tan(x) * tan(eps)))), -tan(x));
	} else if (eps <= 2.45e-6) {
		tmp = (t_2 / (1.0 - (t_2 * (sin(x) / cos(x))))) + (fma(-pow(eps, 4.0), ((t_1 / (t_6 / t_3)) + (sin(x) / (cos(x) / (t_4 - t_0)))), (eps / (t_6 / t_1))) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) + ((t_0 - t_4) * pow(eps, 3.0))));
	} else {
		tmp = (t_5 / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sin(eps) / cos(eps))
	t_3 = Float64(Float64(sin(x) * -0.3333333333333333) / cos(x))
	t_4 = Float64(sin(x) / Float64(cos(x) / t_3))
	t_5 = Float64(tan(x) + tan(eps))
	t_6 = cos(x) ^ 2.0
	tmp = 0.0
	if (eps <= -0.0305)
		tmp = fma(t_5, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), Float64(-tan(x)));
	elseif (eps <= 2.45e-6)
		tmp = Float64(Float64(t_2 / Float64(1.0 - Float64(t_2 * Float64(sin(x) / cos(x))))) + Float64(fma(Float64(-(eps ^ 4.0)), Float64(Float64(t_1 / Float64(t_6 / t_3)) + Float64(sin(x) / Float64(cos(x) / Float64(t_4 - t_0)))), Float64(eps / Float64(t_6 / t_1))) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) + Float64(Float64(t_0 - t_4) * (eps ^ 3.0)))));
	else
		tmp = Float64(Float64(t_5 / Float64(1.0 - Float64(Float64(sin(eps) * sin(x)) / Float64(cos(eps) * cos(x))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -0.0305], N[(t$95$5 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.45e-6], N[(N[(t$95$2 / N[(1.0 - N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[((-N[Power[eps, 4.0], $MachinePrecision]) * N[(N[(t$95$1 / N[(t$95$6 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(t$95$4 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps / N[(t$95$6 / t$95$1), $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[(t$95$0 - t$95$4), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.030499999999999999

   1. Initial program 57.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -0.030499999999999999 < eps < 2.44999999999999984e-6

   1. Initial program 28.8%

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

      1. tan-sum 30.4%

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

      2. div-inv 30.3%

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

      3. fma-neg 30.3%

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

   3. Applied egg-rr 30.3%

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

      1. fma-neg 30.3%

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

      2. associate-*r/ 30.4%

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

      3. *-rgt-identity 30.4%

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

   5. Simplified 30.4%

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

   6. Taylor expanded in x around inf 30.3%

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

      1. associate--l+ 58.4%

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

      2. associate-/r* 58.4%

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

      3. *-commutative 58.4%

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

      4. times-frac 58.4%

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

   8. Simplified 58.4%

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

   9. Taylor expanded in eps around 0 99.7%

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

   11. Simplified 99.8%

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

   if 2.44999999999999984e-6 < eps

   1. Initial program 56.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. 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/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0305:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\mathsf{fma}\left(-{\varepsilon}^{4}, \frac{{\sin x}^{2}}{\frac{{\cos x}^{2}}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}} + \frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}}}, \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\right) + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}}\right) \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.75 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.8e-7) (not (<= eps 2.75e-7)))
   (-
    (/
     (+ (tan x) (tan eps))
     (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
    (tan x))
   (+
    (+ 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)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-7) || !(eps <= 2.75e-7)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	} else {
		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))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.8d-7)) .or. (.not. (eps <= 2.75d-7))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x)
    else
        tmp = (eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-7) || !(eps <= 2.75e-7)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - ((Math.sin(eps) * Math.sin(x)) / (Math.cos(eps) * Math.cos(x))))) - Math.tan(x);
	} else {
		tmp = (eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + ((eps * eps) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.8e-7) or not (eps <= 2.75e-7):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - ((math.sin(eps) * math.sin(x)) / (math.cos(eps) * math.cos(x))))) - math.tan(x)
	else:
		tmp = (eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + ((eps * eps) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.8e-7) || !(eps <= 2.75e-7))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(Float64(sin(eps) * sin(x)) / Float64(cos(eps) * cos(x))))) - tan(x));
	else
		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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.8e-7) || ~((eps <= 2.75e-7)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	else
		tmp = (eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.8e-7], N[Not[LessEqual[eps, 2.75e-7]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(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]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.80000000000000015e-7 or 2.7500000000000001e-7 < eps

   1. Initial program 56.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.3%

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

      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. Taylor expanded in x around inf 99.5%

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

   if -3.80000000000000015e-7 < eps < 2.7500000000000001e-7

   1. Initial program 28.5%

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

      1. tan-sum 29.3%

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

      2. div-inv 29.3%

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

      3. fma-neg 29.4%

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

   3. Applied egg-rr 29.4%

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

      1. fma-neg 29.3%

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

      2. associate-*r/ 29.3%

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

      3. *-rgt-identity 29.3%

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

   5. Simplified 29.3%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

   8. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.75 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (or (<= eps -4.9e-9) (not (<= eps 5.4e-9)))
     (-
      (/
       (+ (tan x) (tan eps))
       (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
      (tan x))
     (+
      (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
      (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if ((eps <= -4.9e-9) || !(eps <= 5.4e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	} else {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + ((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) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    if ((eps <= (-4.9d-9)) .or. (.not. (eps <= 5.4d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x)
    else
        tmp = (t_0 / (1.0d0 - (t_0 * (sin(x) / cos(x))))) + ((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 t_0 = Math.sin(eps) / Math.cos(eps);
	double tmp;
	if ((eps <= -4.9e-9) || !(eps <= 5.4e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - ((Math.sin(eps) * Math.sin(x)) / (Math.cos(eps) * Math.cos(x))))) - Math.tan(x);
	} else {
		tmp = (t_0 / (1.0 - (t_0 * (Math.sin(x) / Math.cos(x))))) + ((eps * Math.pow(Math.sin(x), 2.0)) / Math.pow(Math.cos(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) / math.cos(eps)
	tmp = 0
	if (eps <= -4.9e-9) or not (eps <= 5.4e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - ((math.sin(eps) * math.sin(x)) / (math.cos(eps) * math.cos(x))))) - math.tan(x)
	else:
		tmp = (t_0 / (1.0 - (t_0 * (math.sin(x) / math.cos(x))))) + ((eps * math.pow(math.sin(x), 2.0)) / math.pow(math.cos(x), 2.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if ((eps <= -4.9e-9) || !(eps <= 5.4e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(Float64(sin(eps) * sin(x)) / Float64(cos(eps) * cos(x))))) - tan(x));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) / cos(eps);
	tmp = 0.0;
	if ((eps <= -4.9e-9) || ~((eps <= 5.4e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	else
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + ((eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -4.9e-9], N[Not[LessEqual[eps, 5.4e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.90000000000000004e-9 or 5.4000000000000004e-9 < eps

   1. Initial program 56.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.3%

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

      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. Taylor expanded in x around inf 99.5%

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

   if -4.90000000000000004e-9 < eps < 5.4000000000000004e-9

   1. Initial program 28.5%

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

      1. tan-sum 29.3%

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

      2. div-inv 29.3%

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

      3. fma-neg 29.4%

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

   3. Applied egg-rr 29.4%

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

      1. fma-neg 29.3%

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

      2. associate-*r/ 29.3%

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

      3. *-rgt-identity 29.3%

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

   5. Simplified 29.3%

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

   6. Taylor expanded in x around inf 29.3%

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

      1. associate--l+ 57.9%

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

      2. associate-/r* 57.9%

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

      3. *-commutative 57.9%

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

      4. times-frac 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.15 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.15 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.15e-9) (not (<= eps 3.15e-9)))
   (-
    (/
     (+ (tan x) (tan eps))
     (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
    (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.15e-9) || !(eps <= 3.15e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(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 <= (-3.15d-9)) .or. (.not. (eps <= 3.15d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.15e-9) || !(eps <= 3.15e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - ((Math.sin(eps) * Math.sin(x)) / (Math.cos(eps) * Math.cos(x))))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.15e-9) or not (eps <= 3.15e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - ((math.sin(eps) * math.sin(x)) / (math.cos(eps) * math.cos(x))))) - math.tan(x)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.15e-9) || !(eps <= 3.15e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(Float64(sin(eps) * sin(x)) / Float64(cos(eps) * cos(x))))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.15e-9) || ~((eps <= 3.15e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.15e-9], N[Not[LessEqual[eps, 3.15e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.1500000000000001e-9 or 3.1500000000000001e-9 < eps

   1. Initial program 56.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.3%

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

      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. Taylor expanded in x around inf 99.5%

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

   if -3.1500000000000001e-9 < eps < 3.1500000000000001e-9

   1. Initial program 28.5%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 57.9%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 57.9%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 57.9%

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

      5. frac-times 57.9%

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

      6. tan-quot 57.9%

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

      7. tan-quot 57.9%

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

      8. pow2 57.9%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 57.9%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.15 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.15 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.4e-9) (not (<= eps 4.2e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.4e-9) || !(eps <= 4.2e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(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 <= (-3.4d-9)) .or. (.not. (eps <= 4.2d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.4e-9) || !(eps <= 4.2e-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.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.4e-9) or not (eps <= 4.2e-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.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.4e-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 = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.4e-9) || ~((eps <= 4.2e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.4e-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[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.3999999999999998e-9 or 4.20000000000000039e-9 < eps

   1. Initial program 56.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.3%

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

      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%

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

   if -3.3999999999999998e-9 < eps < 4.20000000000000039e-9

   1. Initial program 28.5%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 57.9%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 57.9%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 57.9%

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

      5. frac-times 57.9%

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

      6. tan-quot 57.9%

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

      7. tan-quot 57.9%

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

      8. pow2 57.9%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 57.9%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 6: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.8e-5)
   (tan eps)
   (if (<= eps 1.15e-6) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5.8e-5) {
		tmp = tan(eps);
	} else if (eps <= 1.15e-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 <= (-5.8d-5)) then
        tmp = tan(eps)
    else if (eps <= 1.15d-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 <= -5.8e-5) {
		tmp = Math.tan(eps);
	} else if (eps <= 1.15e-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 <= -5.8e-5:
		tmp = math.tan(eps)
	elif eps <= 1.15e-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 <= -5.8e-5)
		tmp = tan(eps);
	elseif (eps <= 1.15e-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 <= -5.8e-5)
		tmp = tan(eps);
	elseif (eps <= 1.15e-6)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5.8e-5], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1.15e-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 < -5.8e-5 or 1.15e-6 < eps

   1. Initial program 57.1%

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

   2. Taylor expanded in x around 0 60.3%

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

      1. tan-quot 60.5%

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

      2. expm1-log1p-u 44.9%

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

      3. expm1-udef 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. expm1-def 44.9%

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

      2. expm1-log1p 60.5%

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

   6. Simplified 60.5%

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

   if -5.8e-5 < eps < 1.15e-6

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 98.8%

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

      1. cancel-sign-sub-inv 98.8%

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

      2. metadata-eval 98.8%

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

      3. *-lft-identity 98.8%

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

      4. distribute-lft-in 98.9%

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

      5. *-rgt-identity 98.9%

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

   4. Simplified 98.9%

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

      1. expm1-log1p-u 98.9%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 57.8%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 57.8%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 57.8%

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

      5. frac-times 57.8%

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

      6. tan-quot 57.8%

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

      7. tan-quot 57.8%

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

      8. pow2 57.8%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 57.8%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.0%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7: 58.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

2. Taylor expanded in x around 0 58.7%

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

   1. tan-quot 58.8%

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

   2. expm1-log1p-u 50.8%

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

   3. expm1-udef 25.7%

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

4. Applied egg-rr 25.7%

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

   1. expm1-def 50.8%

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

   2. expm1-log1p 58.8%

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

6. Simplified 58.8%

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

7. Final simplification 58.8%

   \[\leadsto \tan \varepsilon \]

Alternative 8: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 43.0%

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

   1. tan-quot 42.8%

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

   2. clear-num 42.7%

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

3. Applied egg-rr 42.7%

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

   1. add-cube-cbrt 41.8%

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

   2. pow3 41.8%

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

   3. clear-num 41.8%

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

   4. quot-tan 41.8%

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

5. Applied egg-rr 41.8%

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

6. Taylor expanded in eps around 0 4.2%

   \[\leadsto \color{blue}{\frac{\sin x}{\cos x} \cdot {1}^{0.3333333333333333} - \frac{\sin x}{\cos x}} \]
7. Step-by-step derivation

   1. pow-base-1 4.2%

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

   2. *-rgt-identity 4.2%

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

   3. +-inverses 4.2%

      \[\leadsto \color{blue}{0} \]

8. Simplified 4.2%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 4.2%

   \[\leadsto 0 \]

Alternative 9: 31.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 43.0%

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

2. Taylor expanded in x around 0 58.7%

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

3. Taylor expanded in eps around 0 29.9%

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

4. Final simplification 29.9%

   \[\leadsto \varepsilon \]

Developer target: 76.4% 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 2023243 
(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)))