2tan (problem 3.3.2)

Percentage Accurate: 42.5% → 99.3%

Time: 15.8s

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: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}\\ t_1 := \tan \varepsilon \cdot \tan x\\ t_2 := 1 - t_1\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;t_0 + \left(\frac{\tan x}{t_2} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{\frac{\sin x}{\tan x} + \cos x \cdot \left(t_1 + -1\right)}{\frac{\cos x \cdot t_2}{\tan x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/ (tan eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x))))))
        (t_1 (* (tan eps) (tan x)))
        (t_2 (- 1.0 t_1)))
   (if (<= eps -2.9e-9)
     (+ t_0 (- (/ (tan x) t_2) (tan x)))
     (if (<= eps 4.2e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (+
        t_0
        (/
         (+ (/ (sin x) (tan x)) (* (cos x) (+ t_1 -1.0)))
         (/ (* (cos x) t_2) (tan x))))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))));
	double t_1 = tan(eps) * tan(x);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (eps <= -2.9e-9) {
		tmp = t_0 + ((tan(x) / t_2) - tan(x));
	} else if (eps <= 4.2e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = t_0 + (((sin(x) / tan(x)) + (cos(x) * (t_1 + -1.0))) / ((cos(x) * t_2) / tan(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = tan(eps) / (1.0d0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))))
    t_1 = tan(eps) * tan(x)
    t_2 = 1.0d0 - t_1
    if (eps <= (-2.9d-9)) then
        tmp = t_0 + ((tan(x) / t_2) - tan(x))
    else if (eps <= 4.2d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = t_0 + (((sin(x) / tan(x)) + (cos(x) * (t_1 + (-1.0d0)))) / ((cos(x) * t_2) / tan(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) / (1.0 - ((Math.sin(eps) / Math.cos(eps)) * (Math.sin(x) / Math.cos(x))));
	double t_1 = Math.tan(eps) * Math.tan(x);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (eps <= -2.9e-9) {
		tmp = t_0 + ((Math.tan(x) / t_2) - Math.tan(x));
	} else if (eps <= 4.2e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = t_0 + (((Math.sin(x) / Math.tan(x)) + (Math.cos(x) * (t_1 + -1.0))) / ((Math.cos(x) * t_2) / Math.tan(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) / (1.0 - ((math.sin(eps) / math.cos(eps)) * (math.sin(x) / math.cos(x))))
	t_1 = math.tan(eps) * math.tan(x)
	t_2 = 1.0 - t_1
	tmp = 0
	if eps <= -2.9e-9:
		tmp = t_0 + ((math.tan(x) / t_2) - math.tan(x))
	elif eps <= 4.2e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = t_0 + (((math.sin(x) / math.tan(x)) + (math.cos(x) * (t_1 + -1.0))) / ((math.cos(x) * t_2) / math.tan(x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x)))))
	t_1 = Float64(tan(eps) * tan(x))
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (eps <= -2.9e-9)
		tmp = Float64(t_0 + Float64(Float64(tan(x) / t_2) - tan(x)));
	elseif (eps <= 4.2e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(sin(x) / tan(x)) + Float64(cos(x) * Float64(t_1 + -1.0))) / Float64(Float64(cos(x) * t_2) / tan(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))));
	t_1 = tan(eps) * tan(x);
	t_2 = 1.0 - t_1;
	tmp = 0.0;
	if (eps <= -2.9e-9)
		tmp = t_0 + ((tan(x) / t_2) - tan(x));
	elseif (eps <= 4.2e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = t_0 + (((sin(x) / tan(x)) + (cos(x) * (t_1 + -1.0))) / ((cos(x) * t_2) / tan(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[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]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[eps, -2.9e-9], N[(t$95$0 + N[(N[(N[Tan[x], $MachinePrecision] / t$95$2), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.2e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[(N[Sin[x], $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.89999999999999991e-9

   1. Initial program 59.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 98.9%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.0%

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

      1. tan-quot 99.3%

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

      2. expm1-log1p-u 74.4%

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

      3. expm1-udef 73.6%

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

   10. Applied egg-rr 73.6%

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

       1. expm1-def 74.4%

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

       2. expm1-log1p 99.3%

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

   12. Simplified 99.3%

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

       1. tan-quot 99.3%

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

       2. sub-neg 99.3%

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

       3. tan-quot 99.3%

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

       4. tan-quot 99.4%

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

       5. tan-quot 99.4%

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

   14. Applied egg-rr 99.4%

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

       1. sub-neg 99.4%

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

       2. *-commutative 99.4%

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

   16. Simplified 99.4%

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

   if -2.89999999999999991e-9 < eps < 4.20000000000000039e-9

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. *-rgt-identity 99.6%

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

   4. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. frac-times 55.1%

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

      6. tan-quot 55.1%

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

      7. tan-quot 55.1%

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

      8. pow2 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 4.20000000000000039e-9 < eps

   1. Initial program 58.5%

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

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

      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/ 99.5%

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

      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 99.3%

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

      1. associate--l+ 99.3%

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

   8. Simplified 99.3%

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

      1. tan-quot 99.5%

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

      2. expm1-log1p-u 78.7%

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

      3. expm1-udef 78.2%

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

   10. Applied egg-rr 78.2%

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

       1. expm1-def 78.7%

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

       2. expm1-log1p 99.5%

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

   12. Simplified 99.5%

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

       1. associate-/l/ 99.4%

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

       2. clear-num 99.4%

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

       3. frac-sub 99.5%

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

   14. Applied egg-rr 99.6%

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

       1. associate-*r/ 99.6%

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

       2. *-rgt-identity 99.6%

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

       3. *-rgt-identity 99.6%

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

       4. *-commutative 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-*r/ 99.6%

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

   16. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\frac{\sin x}{\tan x} + \cos x \cdot \left(\tan \varepsilon \cdot \tan x + -1\right)}{\frac{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\tan x}}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -4e-9)
     (+
      (/ (tan eps) (- 1.0 (* (/ (sin eps) (cos eps)) (/ (sin x) (cos x)))))
      (- (/ (tan x) t_0) (tan x)))
     (if (<= eps 2.1e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (/ (+ (tan eps) (tan x)) t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -4e-9) {
		tmp = (tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))))) + ((tan(x) / t_0) - tan(x));
	} else if (eps <= 2.1e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = ((tan(eps) + tan(x)) / t_0) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (tan(eps) * tan(x))
    if (eps <= (-4d-9)) then
        tmp = (tan(eps) / (1.0d0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))))) + ((tan(x) / t_0) - tan(x))
    else if (eps <= 2.1d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = ((tan(eps) + tan(x)) / t_0) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double tmp;
	if (eps <= -4e-9) {
		tmp = (Math.tan(eps) / (1.0 - ((Math.sin(eps) / Math.cos(eps)) * (Math.sin(x) / Math.cos(x))))) + ((Math.tan(x) / t_0) - Math.tan(x));
	} else if (eps <= 2.1e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = ((Math.tan(eps) + Math.tan(x)) / t_0) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 - (math.tan(eps) * math.tan(x))
	tmp = 0
	if eps <= -4e-9:
		tmp = (math.tan(eps) / (1.0 - ((math.sin(eps) / math.cos(eps)) * (math.sin(x) / math.cos(x))))) + ((math.tan(x) / t_0) - math.tan(x))
	elif eps <= 2.1e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = ((math.tan(eps) + math.tan(x)) / t_0) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -4e-9)
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(eps) / cos(eps)) * Float64(sin(x) / cos(x))))) + Float64(Float64(tan(x) / t_0) - tan(x)));
	elseif (eps <= 2.1e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / t_0) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 - (tan(eps) * tan(x));
	tmp = 0.0;
	if (eps <= -4e-9)
		tmp = (tan(eps) / (1.0 - ((sin(eps) / cos(eps)) * (sin(x) / cos(x))))) + ((tan(x) / t_0) - tan(x));
	elseif (eps <= 2.1e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = ((tan(eps) + tan(x)) / t_0) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4e-9], N[(N[(N[Tan[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] + N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.1e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.00000000000000025e-9

   1. Initial program 59.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 98.9%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.0%

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

      1. tan-quot 99.3%

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

      2. expm1-log1p-u 74.4%

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

      3. expm1-udef 73.6%

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

   10. Applied egg-rr 73.6%

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

       1. expm1-def 74.4%

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

       2. expm1-log1p 99.3%

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

   12. Simplified 99.3%

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

       1. tan-quot 99.3%

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

       2. sub-neg 99.3%

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

       3. tan-quot 99.3%

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

       4. tan-quot 99.4%

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

       5. tan-quot 99.4%

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

   14. Applied egg-rr 99.4%

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

       1. sub-neg 99.4%

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

       2. *-commutative 99.4%

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

   16. Simplified 99.4%

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

   if -4.00000000000000025e-9 < eps < 2.10000000000000019e-9

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. *-rgt-identity 99.6%

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

   4. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. frac-times 55.1%

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

      6. tan-quot 55.1%

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

      7. tan-quot 55.1%

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

      8. pow2 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 2.10000000000000019e-9 < eps

   1. Initial program 58.5%

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

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

      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/ 99.5%

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

      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%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 + \left(-1 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -2.6e-9)
     (- (/ t_0 (+ 1.0 (- -1.0 (fma (tan x) (tan eps) -1.0)))) (tan x))
     (if (<= eps 3.1e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (/ t_0 (- 1.0 (* (tan eps) (tan x)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -2.6e-9) {
		tmp = (t_0 / (1.0 + (-1.0 - fma(tan(x), tan(eps), -1.0)))) - tan(x);
	} else if (eps <= 3.1e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -2.6e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 + Float64(-1.0 - fma(tan(x), tan(eps), -1.0)))) - tan(x));
	elseif (eps <= 3.1e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.6e-9], N[(N[(t$95$0 / N[(1.0 + N[(-1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.1e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.6000000000000001e-9

   1. Initial program 59.4%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 91.5%

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

      3. log1p-udef 91.5%

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

      4. add-exp-log 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. associate--l+ 99.4%

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

      2. fma-neg 99.4%

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

      3. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   if -2.6000000000000001e-9 < eps < 3.10000000000000005e-9

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. *-rgt-identity 99.6%

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

   4. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. frac-times 55.1%

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

      6. tan-quot 55.1%

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

      7. tan-quot 55.1%

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

      8. pow2 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 3.10000000000000005e-9 < eps

   1. Initial program 58.5%

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

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

      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/ 99.5%

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

      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%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 + \left(-1 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.3e-9) (not (<= eps 3.5e-9)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-9) || !(eps <= 3.5e-9)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(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 <= (-3.3d-9)) .or. (.not. (eps <= 3.5d-9))) then
        tmp = ((tan(eps) + tan(x)) / (1.0d0 - (tan(eps) * tan(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 <= -3.3e-9) || !(eps <= 3.5e-9)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / (1.0 - (Math.tan(eps) * Math.tan(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 <= -3.3e-9) or not (eps <= 3.5e-9):
		tmp = ((math.tan(eps) + math.tan(x)) / (1.0 - (math.tan(eps) * math.tan(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 <= -3.3e-9) || !(eps <= 3.5e-9))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(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 <= -3.3e-9) || ~((eps <= 3.5e-9)))
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.3e-9], N[Not[LessEqual[eps, 3.5e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $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 < -3.30000000000000018e-9 or 3.4999999999999999e-9 < eps

   1. Initial program 58.9%

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

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

      3. fma-neg 99.4%

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

   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 99.3%

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

      2. associate-*r/ 99.5%

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

      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%

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

   if -3.30000000000000018e-9 < eps < 3.4999999999999999e-9

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. *-rgt-identity 99.6%

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

   4. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. frac-times 55.1%

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

      6. tan-quot 55.1%

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

      7. tan-quot 55.1%

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

      8. pow2 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 5: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00039:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.4e-5)
   (- (/ 1.0 (/ (cos eps) (sin eps))) (tan x))
   (if (<= eps 0.00039) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.4e-5) {
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	} else if (eps <= 0.00039) {
		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 <= (-2.4d-5)) then
        tmp = (1.0d0 / (cos(eps) / sin(eps))) - tan(x)
    else if (eps <= 0.00039d0) 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 <= -2.4e-5) {
		tmp = (1.0 / (Math.cos(eps) / Math.sin(eps))) - Math.tan(x);
	} else if (eps <= 0.00039) {
		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 <= -2.4e-5:
		tmp = (1.0 / (math.cos(eps) / math.sin(eps))) - math.tan(x)
	elif eps <= 0.00039:
		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 <= -2.4e-5)
		tmp = Float64(Float64(1.0 / Float64(cos(eps) / sin(eps))) - tan(x));
	elseif (eps <= 0.00039)
		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 <= -2.4e-5)
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	elseif (eps <= 0.00039)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.4e-5], N[(N[(1.0 / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00039], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.4000000000000001e-5

   1. Initial program 59.4%

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

      1. tan-sum 99.4%

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

      2. clear-num 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x around 0 62.1%

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

   if -2.4000000000000001e-5 < eps < 3.89999999999999993e-4

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. *-rgt-identity 99.6%

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

   4. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. frac-times 55.1%

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

      6. tan-quot 55.1%

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

      7. tan-quot 55.1%

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

      8. pow2 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 3.89999999999999993e-4 < eps

   1. Initial program 58.5%

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

   2. Taylor expanded in x around 0 59.9%

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

      1. tan-quot 99.5%

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

      2. expm1-log1p-u 78.7%

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

      3. expm1-udef 78.2%

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

   4. Applied egg-rr 48.8%

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

      1. expm1-def 78.7%

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

      2. expm1-log1p 99.5%

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

   6. Simplified 60.1%

      \[\leadsto \color{blue}{\tan \varepsilon} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00039:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 6: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -8.5e-7)
   (tan eps)
   (if (<= eps 2.1e-5) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5e-7) {
		tmp = tan(eps);
	} else if (eps <= 2.1e-5) {
		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 <= (-8.5d-7)) then
        tmp = tan(eps)
    else if (eps <= 2.1d-5) 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 <= -8.5e-7) {
		tmp = Math.tan(eps);
	} else if (eps <= 2.1e-5) {
		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 <= -8.5e-7:
		tmp = math.tan(eps)
	elif eps <= 2.1e-5:
		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 <= -8.5e-7)
		tmp = tan(eps);
	elseif (eps <= 2.1e-5)
		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 <= -8.5e-7)
		tmp = tan(eps);
	elseif (eps <= 2.1e-5)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -8.5e-7], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 2.1e-5], 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 < -8.50000000000000014e-7 or 2.09999999999999988e-5 < eps

   1. Initial program 58.9%

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

   2. Taylor expanded in x around 0 60.6%

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

      1. tan-quot 99.4%

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

      2. expm1-log1p-u 76.7%

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

      3. expm1-udef 76.1%

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

   4. Applied egg-rr 48.7%

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

      1. expm1-def 76.7%

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

      2. expm1-log1p 99.4%

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

   6. Simplified 60.8%

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

   if -8.50000000000000014e-7 < eps < 2.09999999999999988e-5

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. *-rgt-identity 99.6%

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

   4. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. frac-times 55.1%

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

      6. tan-quot 55.1%

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

      7. tan-quot 55.1%

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

      8. pow2 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7: 34.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.5%

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

2. Taylor expanded in x around 0 58.1%

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

   1. add-sqr-sqrt 31.3%

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

   2. sqrt-unprod 27.2%

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

   3. pow2 27.2%

      \[\leadsto \frac{\sqrt{\color{blue}{{\sin \varepsilon}^{2}}}}{\cos \varepsilon} \]

4. Applied egg-rr 27.2%

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

   1. unpow2 27.2%

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

   2. rem-sqrt-square 33.3%

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

6. Simplified 33.3%

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

7. Taylor expanded in eps around 0 22.3%

   \[\leadsto \color{blue}{\left|\sin \varepsilon\right|} \]
8. Step-by-step derivation

   1. rem-square-sqrt 18.2%

      \[\leadsto \left|\color{blue}{\sqrt{\sin \varepsilon} \cdot \sqrt{\sin \varepsilon}}\right| \]

   2. fabs-sqr 18.2%

      \[\leadsto \color{blue}{\sqrt{\sin \varepsilon} \cdot \sqrt{\sin \varepsilon}} \]

   3. rem-square-sqrt 31.5%

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

9. Simplified 31.5%

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

10. Final simplification 31.5%

    \[\leadsto \sin \varepsilon \]

Alternative 8: 58.1% 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 45.5%

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

2. Taylor expanded in x around 0 58.1%

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

   1. tan-quot 78.8%

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

   2. expm1-log1p-u 66.7%

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

   3. expm1-udef 43.7%

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

4. Applied egg-rr 29.1%

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

   1. expm1-def 66.7%

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

   2. expm1-log1p 78.8%

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

6. Simplified 58.2%

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

7. Final simplification 58.2%

   \[\leadsto \tan \varepsilon \]

Alternative 9: 31.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 45.5%

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

2. Taylor expanded in x around 0 58.1%

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

3. Taylor expanded in eps around 0 27.9%

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

4. Final simplification 27.9%

   \[\leadsto \varepsilon \]

Developer target: 76.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023242 
(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)))