2tan (problem 3.3.2)

Percentage Accurate: 43.0% → 99.5%

Time: 20.4s

Alternatives: 12

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: 43.0% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -4.7e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 4.1e-7)
       (/ (/ eps (/ (cos x) (fma (sin x) (tan x) (cos x)))) t_1)
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -4.7e-7) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 4.1e-7) {
		tmp = (eps / (cos(x) / fma(sin(x), tan(x), cos(x)))) / t_1;
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -4.7e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 4.1e-7)
		tmp = Float64(Float64(eps / Float64(cos(x) / fma(sin(x), tan(x), cos(x)))) / t_1);
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.7e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.1e-7], N[(N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.7e-7

   1. Initial program 58.9%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. tan-quot 99.4%

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

      3. associate-*r/ 99.4%

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

   8. Applied egg-rr 99.4%

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

   if -4.7e-7 < eps < 4.0999999999999999e-7

   1. Initial program 29.6%

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

      1. tan-sum 31.1%

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

      2. tan-quot 30.7%

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

      3. frac-sub 30.6%

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

   4. Applied egg-rr 30.6%

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

   5. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   7. Simplified 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. tan-quot 99.5%

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

      3. tan-quot 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.5%

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

      6. *-commutative 99.5%

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

      7. tan-quot 99.5%

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

      8. tan-quot 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-*l/ 99.6%

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

       2. *-lft-identity 99.6%

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

       3. associate-/l* 99.6%

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

       4. +-commutative 99.6%

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

       5. /-rgt-identity 99.6%

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

       6. fma-def 99.7%

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

   11. Simplified 99.7%

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

   if 4.0999999999999999e-7 < eps

   1. Initial program 59.5%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

      8. *-un-lft-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. fma-neg 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}}{1 - \tan x \cdot \tan \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \frac{t_0}{\frac{-1 + t_0}{\tan x}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))))
   (-
    (/ (/ (sin eps) (cos eps)) (- 1.0 (/ (tan x) (/ 1.0 (tan eps)))))
    (/ t_0 (/ (+ -1.0 t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	return ((sin(eps) / cos(eps)) / (1.0 - (tan(x) / (1.0 / tan(eps))))) - (t_0 / ((-1.0 + t_0) / tan(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(x) * tan(eps)
    code = ((sin(eps) / cos(eps)) / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - (t_0 / (((-1.0d0) + t_0) / tan(x)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) * Math.tan(eps);
	return ((Math.sin(eps) / Math.cos(eps)) / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - (t_0 / ((-1.0 + t_0) / Math.tan(x)));
}

Python

def code(x, eps):
	t_0 = math.tan(x) * math.tan(eps)
	return ((math.sin(eps) / math.cos(eps)) / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - (t_0 / ((-1.0 + t_0) / math.tan(x)))

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	return Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - Float64(t_0 / Float64(Float64(-1.0 + t_0) / tan(x))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) * tan(eps);
	tmp = ((sin(eps) / cos(eps)) / (1.0 - (tan(x) / (1.0 / tan(eps))))) - (t_0 / ((-1.0 + t_0) / tan(x)));
end

Wolfram

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

Derivation

1. Initial program 45.0%

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

   1. tan-sum 66.6%

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

   2. div-inv 66.6%

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

   3. *-un-lft-identity 66.6%

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

   4. prod-diff 66.6%

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

   5. *-commutative 66.6%

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

   6. *-un-lft-identity 66.6%

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

   7. *-commutative 66.6%

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

   8. *-un-lft-identity 66.6%

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

4. Applied egg-rr 66.6%

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

   1. +-commutative 66.6%

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

   2. fma-udef 66.6%

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

   3. associate-+r+ 66.6%

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

   4. unsub-neg 66.6%

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

6. Simplified 66.6%

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

7. Taylor expanded in x around inf 66.4%

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

   1. associate--l+ 80.5%

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

   2. associate-/r* 80.5%

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

   3. times-frac 80.5%

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

9. Simplified 80.6%

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

    1. frac-2neg 80.6%

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

    2. clear-num 77.6%

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

    3. frac-sub 76.6%

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

11. Applied egg-rr 77.4%

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

    1. cancel-sign-sub 77.4%

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

    2. distribute-lft-neg-out 77.4%

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

    3. rgt-mult-inverse 80.6%

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

    4. metadata-eval 80.6%

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

    5. *-rgt-identity 80.6%

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

    6. sub-neg 80.6%

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

    7. distribute-rgt-neg-out 80.6%

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

    8. associate-+r+ 99.5%

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

    9. metadata-eval 99.5%

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

    10. associate-*r/ 99.5%

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

13. Simplified 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \color{blue}{\frac{0 + \tan x \cdot \left(-\tan \varepsilon\right)}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x}}} \]
14. Step-by-step derivation

    1. *-commutative 99.5%

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

    2. clear-num 99.5%

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

    3. un-div-inv 99.5%

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

    4. tan-quot 99.6%

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

    5. clear-num 99.6%

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

    6. tan-quot 99.5%

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

15. Applied egg-rr 99.5%

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

16. Final simplification 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \frac{\tan x \cdot \tan \varepsilon}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x}} \]
17. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - t_0} - \frac{t_0}{\frac{-1 + t_0}{\tan x}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))))
   (-
    (/ (/ (sin eps) (cos eps)) (- 1.0 t_0))
    (/ t_0 (/ (+ -1.0 t_0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	return ((sin(eps) / cos(eps)) / (1.0 - t_0)) - (t_0 / ((-1.0 + t_0) / tan(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(x) * tan(eps)
    code = ((sin(eps) / cos(eps)) / (1.0d0 - t_0)) - (t_0 / (((-1.0d0) + t_0) / tan(x)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) * Math.tan(eps);
	return ((Math.sin(eps) / Math.cos(eps)) / (1.0 - t_0)) - (t_0 / ((-1.0 + t_0) / Math.tan(x)));
}

Python

def code(x, eps):
	t_0 = math.tan(x) * math.tan(eps)
	return ((math.sin(eps) / math.cos(eps)) / (1.0 - t_0)) - (t_0 / ((-1.0 + t_0) / math.tan(x)))

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	return Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - t_0)) - Float64(t_0 / Float64(Float64(-1.0 + t_0) / tan(x))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) * tan(eps);
	tmp = ((sin(eps) / cos(eps)) / (1.0 - t_0)) - (t_0 / ((-1.0 + t_0) / tan(x)));
end

Wolfram

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

Derivation

1. Initial program 45.0%

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

   1. tan-sum 66.6%

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

   2. div-inv 66.6%

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

   3. *-un-lft-identity 66.6%

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

   4. prod-diff 66.6%

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

   5. *-commutative 66.6%

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

   6. *-un-lft-identity 66.6%

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

   7. *-commutative 66.6%

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

   8. *-un-lft-identity 66.6%

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

4. Applied egg-rr 66.6%

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

   1. +-commutative 66.6%

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

   2. fma-udef 66.6%

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

   3. associate-+r+ 66.6%

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

   4. unsub-neg 66.6%

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

6. Simplified 66.6%

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

7. Taylor expanded in x around inf 66.4%

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

   1. associate--l+ 80.5%

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

   2. associate-/r* 80.5%

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

   3. times-frac 80.5%

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

9. Simplified 80.6%

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

    1. frac-2neg 80.6%

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

    2. clear-num 77.6%

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

    3. frac-sub 76.6%

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

11. Applied egg-rr 77.4%

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

    1. cancel-sign-sub 77.4%

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

    2. distribute-lft-neg-out 77.4%

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

    3. rgt-mult-inverse 80.6%

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

    4. metadata-eval 80.6%

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

    5. *-rgt-identity 80.6%

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

    6. sub-neg 80.6%

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

    7. distribute-rgt-neg-out 80.6%

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

    8. associate-+r+ 99.5%

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

    9. metadata-eval 99.5%

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

    10. associate-*r/ 99.5%

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

13. Simplified 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \color{blue}{\frac{0 + \tan x \cdot \left(-\tan \varepsilon\right)}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x}}} \]
14. Step-by-step derivation

    1. *-commutative 99.5%

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

    2. clear-num 99.5%

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

    3. un-div-inv 99.5%

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

    4. tan-quot 99.6%

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

    5. clear-num 99.6%

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

    6. tan-quot 99.5%

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

15. Applied egg-rr 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\frac{\tan x}{\frac{1}{\tan \varepsilon}}}} + \frac{0 + \tan x \cdot \left(-\tan \varepsilon\right)}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x}} \]
16. Step-by-step derivation

    1. associate-/r/ 99.5%

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

    2. /-rgt-identity 99.5%

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

17. Simplified 99.5%

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

18. Final simplification 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \frac{\tan x \cdot \tan \varepsilon}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x}} \]
19. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon}{t_0} \cdot \frac{\cos x + \tan x \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{t_0}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps)))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -6.2e-7)
     (- (/ t_1 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 4.5e-7)
       (* (/ eps t_0) (/ (+ (cos x) (* (tan x) (sin x))) (cos x)))
       (fma t_1 (/ 1.0 t_0) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -6.2e-7) {
		tmp = (t_1 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 4.5e-7) {
		tmp = (eps / t_0) * ((cos(x) + (tan(x) * sin(x))) / cos(x));
	} else {
		tmp = fma(t_1, (1.0 / t_0), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -6.2e-7)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 4.5e-7)
		tmp = Float64(Float64(eps / t_0) * Float64(Float64(cos(x) + Float64(tan(x) * sin(x))) / cos(x)));
	else
		tmp = fma(t_1, Float64(1.0 / t_0), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.2e-7], N[(N[(t$95$1 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-7], N[(N[(eps / t$95$0), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / t$95$0), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.1999999999999999e-7

   1. Initial program 58.9%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. tan-quot 99.4%

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

      3. associate-*r/ 99.4%

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

   8. Applied egg-rr 99.4%

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

   if -6.1999999999999999e-7 < eps < 4.4999999999999998e-7

   1. Initial program 29.6%

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

      1. tan-sum 31.1%

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

      2. tan-quot 30.7%

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

      3. frac-sub 30.6%

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

   4. Applied egg-rr 30.6%

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

   5. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   7. Simplified 99.5%

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

      1. times-frac 99.5%

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

      2. unpow2 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. times-frac 99.5%

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

      5. tan-quot 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. /-rgt-identity 99.5%

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

   11. Simplified 99.5%

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

   if 4.4999999999999998e-7 < eps

   1. Initial program 59.5%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

      8. *-un-lft-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. fma-neg 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\cos x + \tan x \cdot \sin x}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -4.4e-9)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 3.1e-9)
       (/ eps (/ (cos x) (fma (sin x) (tan x) (cos x))))
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -4.4e-9) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 3.1e-9) {
		tmp = eps / (cos(x) / fma(sin(x), tan(x), cos(x)));
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -4.4e-9)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 3.1e-9)
		tmp = Float64(eps / Float64(cos(x) / fma(sin(x), tan(x), cos(x))));
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.4e-9], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.1e-9], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.3999999999999997e-9

   1. Initial program 58.2%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. prod-diff 98.9%

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

      5. *-commutative 98.9%

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

      6. *-un-lft-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. *-un-lft-identity 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative 98.9%

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

      2. fma-udef 98.9%

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

      3. associate-+r+ 98.9%

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

      4. unsub-neg 98.9%

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

   6. Simplified 98.9%

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

   if -4.3999999999999997e-9 < eps < 3.10000000000000005e-9

   1. Initial program 29.8%

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

      1. tan-sum 30.8%

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

      2. tan-quot 30.4%

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

      3. frac-sub 30.3%

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

   4. Applied egg-rr 30.3%

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

   5. Taylor expanded in eps around 0 98.9%

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

      1. associate-/l* 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-lft-identity 98.9%

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

   7. Simplified 98.9%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 98.2%

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

      3. unpow2 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. times-frac 98.2%

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

      6. tan-quot 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. expm1-def 98.9%

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

       2. expm1-log1p 99.0%

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

       3. +-commutative 99.0%

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

       4. /-rgt-identity 99.0%

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

       5. fma-def 99.1%

          \[\leadsto \frac{\varepsilon}{\frac{\cos x}{\color{blue}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}} \]

   11. Simplified 99.1%

       \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}} \]

   if 3.10000000000000005e-9 < eps

   1. Initial program 59.5%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

      8. *-un-lft-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. fma-neg 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2.7e-9)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 5.2e-9)
       (/ eps (/ (cos x) (fma (sin x) (tan x) (cos x))))
       (fma t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.7e-9) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 5.2e-9) {
		tmp = eps / (cos(x) / fma(sin(x), tan(x), cos(x)));
	} else {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(x) * tan(eps)))), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.7e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 5.2e-9)
		tmp = Float64(eps / Float64(cos(x) / fma(sin(x), tan(x), cos(x))));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.7e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-9], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.7000000000000002e-9

   1. Initial program 58.2%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. prod-diff 98.9%

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

      5. *-commutative 98.9%

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

      6. *-un-lft-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. *-un-lft-identity 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative 98.9%

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

      2. fma-udef 98.9%

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

      3. associate-+r+ 98.9%

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

      4. unsub-neg 98.9%

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

   6. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. tan-quot 98.9%

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

      3. associate-*r/ 98.9%

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

   8. Applied egg-rr 98.9%

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

   if -2.7000000000000002e-9 < eps < 5.2000000000000002e-9

   1. Initial program 29.8%

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

      1. tan-sum 30.8%

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

      2. tan-quot 30.4%

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

      3. frac-sub 30.3%

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

   4. Applied egg-rr 30.3%

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

   5. Taylor expanded in eps around 0 98.9%

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

      1. associate-/l* 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-lft-identity 98.9%

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

   7. Simplified 98.9%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 98.2%

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

      3. unpow2 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. times-frac 98.2%

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

      6. tan-quot 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. expm1-def 98.9%

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

       2. expm1-log1p 99.0%

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

       3. +-commutative 99.0%

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

       4. /-rgt-identity 99.0%

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

       5. fma-def 99.1%

          \[\leadsto \frac{\varepsilon}{\frac{\cos x}{\color{blue}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}} \]

   11. Simplified 99.1%

       \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}} \]

   if 5.2000000000000002e-9 < eps

   1. Initial program 59.5%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. prod-diff 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. *-commutative 99.6%

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

      8. *-un-lft-identity 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-udef 99.6%

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

      3. associate-+r+ 99.6%

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

      4. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. fma-neg 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.35e-9) (not (<= eps 2.3e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (/ eps (/ (cos x) (fma (sin x) (tan x) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.35e-9) || !(eps <= 2.3e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps / (cos(x) / fma(sin(x), tan(x), cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.35e-9) || !(eps <= 2.3e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps / Float64(cos(x) / fma(sin(x), tan(x), cos(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.35e-9], N[Not[LessEqual[eps, 2.3e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.3500000000000001e-9 or 2.2999999999999999e-9 < eps

   1. Initial program 58.8%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.2%

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

      3. associate-+r+ 99.2%

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

      4. unsub-neg 99.2%

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

   6. Simplified 99.2%

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

   if -1.3500000000000001e-9 < eps < 2.2999999999999999e-9

   1. Initial program 29.8%

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

      1. tan-sum 30.8%

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

      2. tan-quot 30.4%

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

      3. frac-sub 30.3%

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

   4. Applied egg-rr 30.3%

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

   5. Taylor expanded in eps around 0 98.9%

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

      1. associate-/l* 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-lft-identity 98.9%

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

   7. Simplified 98.9%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 98.2%

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

      3. unpow2 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. times-frac 98.2%

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

      6. tan-quot 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. expm1-def 98.9%

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

       2. expm1-log1p 99.0%

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

       3. +-commutative 99.0%

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

       4. /-rgt-identity 99.0%

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

       5. fma-def 99.1%

          \[\leadsto \frac{\varepsilon}{\frac{\cos x}{\color{blue}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}} \]

   11. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 1.22 \cdot 10^{-5}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 1.22e-5)))
   (- (tan eps) (tan x))
   (/ eps (/ (cos x) (fma (sin x) (tan x) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 1.22e-5)) {
		tmp = tan(eps) - tan(x);
	} else {
		tmp = eps / (cos(x) / fma(sin(x), tan(x), cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 1.22e-5))
		tmp = Float64(tan(eps) - tan(x));
	else
		tmp = Float64(eps / Float64(cos(x) / fma(sin(x), tan(x), cos(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 1.22e-5]], $MachinePrecision]], N[(N[Tan[eps], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00820000000000000069 or 1.22000000000000001e-5 < eps

   1. Initial program 59.6%

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

      1. expm1-log1p-u 44.9%

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

   4. Applied egg-rr 44.9%

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

   5. Taylor expanded in x around 0 46.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) - \tan x \]
   6. Step-by-step derivation

      1. log1p-def 46.7%

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

   7. Simplified 46.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) - \tan x \]
   8. Step-by-step derivation

      1. expm1-log1p-u 62.0%

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

      2. tan-quot 62.0%

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

      3. sub-neg 62.0%

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

      4. tan-quot 62.3%

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

      5. tan-quot 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. sub-neg 62.3%

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

   11. Simplified 62.3%

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

   if -0.00820000000000000069 < eps < 1.22000000000000001e-5

   1. Initial program 29.4%

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

      1. tan-sum 31.6%

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

      2. tan-quot 31.2%

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

      3. frac-sub 31.1%

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

   4. Applied egg-rr 31.1%

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

   5. Taylor expanded in eps around 0 97.9%

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

      1. associate-/l* 98.0%

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

      2. cancel-sign-sub-inv 98.0%

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

      3. metadata-eval 98.0%

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

      4. *-lft-identity 98.0%

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

   7. Simplified 98.0%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 97.2%

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

      3. unpow2 97.2%

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

      4. *-un-lft-identity 97.2%

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

      5. times-frac 97.2%

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

      6. tan-quot 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. expm1-def 97.9%

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

       2. expm1-log1p 98.0%

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

       3. +-commutative 98.0%

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

       4. /-rgt-identity 98.0%

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

       5. fma-def 98.1%

          \[\leadsto \frac{\varepsilon}{\frac{\cos x}{\color{blue}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}} \]

   11. Simplified 98.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 1.22 \cdot 10^{-5}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\mathsf{fma}\left(\sin x, \tan x, \cos x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.009 \lor \neg \left(\varepsilon \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.009) (not (<= eps 2e-5)))
   (- (tan eps) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.009) || !(eps <= 2e-5)) {
		tmp = tan(eps) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.009d0)) .or. (.not. (eps <= 2d-5))) then
        tmp = tan(eps) - tan(x)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.009) || !(eps <= 2e-5)) {
		tmp = Math.tan(eps) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.009) or not (eps <= 2e-5):
		tmp = math.tan(eps) - math.tan(x)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.009) || !(eps <= 2e-5))
		tmp = Float64(tan(eps) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.009) || ~((eps <= 2e-5)))
		tmp = tan(eps) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.009], N[Not[LessEqual[eps, 2e-5]], $MachinePrecision]], N[(N[Tan[eps], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00899999999999999932 or 2.00000000000000016e-5 < eps

   1. Initial program 59.6%

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

      1. expm1-log1p-u 44.9%

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

   4. Applied egg-rr 44.9%

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

   5. Taylor expanded in x around 0 46.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) - \tan x \]
   6. Step-by-step derivation

      1. log1p-def 46.7%

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

   7. Simplified 46.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) - \tan x \]
   8. Step-by-step derivation

      1. expm1-log1p-u 62.0%

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

      2. tan-quot 62.0%

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

      3. sub-neg 62.0%

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

      4. tan-quot 62.3%

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

      5. tan-quot 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. sub-neg 62.3%

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

   11. Simplified 62.3%

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

   if -0.00899999999999999932 < eps < 2.00000000000000016e-5

   1. Initial program 29.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. metadata-eval 97.9%

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

      3. *-lft-identity 97.9%

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

   5. Simplified 97.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.009 \lor \neg \left(\varepsilon \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \tan x \cdot \sin x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 2.7e-5)))
   (- (tan eps) (tan x))
   (/ eps (/ (cos x) (+ (cos x) (* (tan x) (sin x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 2.7e-5)) {
		tmp = tan(eps) - tan(x);
	} else {
		tmp = eps / (cos(x) / (cos(x) + (tan(x) * sin(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0082d0)) .or. (.not. (eps <= 2.7d-5))) then
        tmp = tan(eps) - tan(x)
    else
        tmp = eps / (cos(x) / (cos(x) + (tan(x) * sin(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 2.7e-5)) {
		tmp = Math.tan(eps) - Math.tan(x);
	} else {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + (Math.tan(x) * Math.sin(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0082) or not (eps <= 2.7e-5):
		tmp = math.tan(eps) - math.tan(x)
	else:
		tmp = eps / (math.cos(x) / (math.cos(x) + (math.tan(x) * math.sin(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 2.7e-5))
		tmp = Float64(tan(eps) - tan(x));
	else
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64(tan(x) * sin(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0082) || ~((eps <= 2.7e-5)))
		tmp = tan(eps) - tan(x);
	else
		tmp = eps / (cos(x) / (cos(x) + (tan(x) * sin(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 2.7e-5]], $MachinePrecision]], N[(N[Tan[eps], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00820000000000000069 or 2.6999999999999999e-5 < eps

   1. Initial program 59.6%

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

      1. expm1-log1p-u 44.9%

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

   4. Applied egg-rr 44.9%

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

   5. Taylor expanded in x around 0 46.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) - \tan x \]
   6. Step-by-step derivation

      1. log1p-def 46.7%

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

   7. Simplified 46.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) - \tan x \]
   8. Step-by-step derivation

      1. expm1-log1p-u 62.0%

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

      2. tan-quot 62.0%

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

      3. sub-neg 62.0%

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

      4. tan-quot 62.3%

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

      5. tan-quot 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. sub-neg 62.3%

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

   11. Simplified 62.3%

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

   if -0.00820000000000000069 < eps < 2.6999999999999999e-5

   1. Initial program 29.4%

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

      1. tan-sum 31.6%

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

      2. tan-quot 31.2%

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

      3. frac-sub 31.1%

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

   4. Applied egg-rr 31.1%

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

   5. Taylor expanded in eps around 0 97.9%

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

      1. associate-/l* 98.0%

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

      2. cancel-sign-sub-inv 98.0%

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

      3. metadata-eval 98.0%

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

      4. *-lft-identity 98.0%

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

   7. Simplified 98.0%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 97.2%

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

      3. unpow2 97.2%

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

      4. *-un-lft-identity 97.2%

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

      5. times-frac 97.2%

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

      6. tan-quot 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. expm1-def 97.9%

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

       2. expm1-log1p 98.0%

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

       3. /-rgt-identity 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \tan x \cdot \sin x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% 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 45.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 60.1%

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

   1. tan-quot 60.3%

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

   2. expm1-log1p-u 52.6%

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

   3. expm1-udef 27.4%

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

5. Applied egg-rr 27.4%

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

   1. expm1-def 52.6%

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

   2. expm1-log1p 60.3%

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

7. Simplified 60.3%

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

8. Final simplification 60.3%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 12: 31.9% 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 45.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 60.1%

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

4. Taylor expanded in eps around 0 30.1%

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

5. Final simplification 30.1%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 77.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(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)))