2tan (problem 3.3.2)

Percentage Accurate: 42.3% → 99.4%

Time: 20.0s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.3% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := \tan \varepsilon \cdot \tan x\\ t_2 := 1 - t_1\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_2} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \tan x \cdot \frac{t_1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x)))
        (t_1 (* (tan eps) (tan x)))
        (t_2 (- 1.0 t_1)))
   (if (<= eps -5.8e-9)
     (- (/ t_0 t_2) (tan x))
     (if (<= eps 6.2e-9)
       (+ eps (* (tan x) (/ t_1 t_2)))
       (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = tan(eps) * tan(x);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (eps <= -5.8e-9) {
		tmp = (t_0 / t_2) - tan(x);
	} else if (eps <= 6.2e-9) {
		tmp = eps + (tan(x) * (t_1 / t_2));
	} else {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = tan(eps) + tan(x)
    t_1 = tan(eps) * tan(x)
    t_2 = 1.0d0 - t_1
    if (eps <= (-5.8d-9)) then
        tmp = (t_0 / t_2) - tan(x)
    else if (eps <= 6.2d-9) then
        tmp = eps + (tan(x) * (t_1 / t_2))
    else
        tmp = (t_0 / (1.0d0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) + Math.tan(x);
	double t_1 = Math.tan(eps) * Math.tan(x);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (eps <= -5.8e-9) {
		tmp = (t_0 / t_2) - Math.tan(x);
	} else if (eps <= 6.2e-9) {
		tmp = eps + (Math.tan(x) * (t_1 / t_2));
	} else {
		tmp = (t_0 / (1.0 - ((Math.tan(x) * Math.sin(eps)) / Math.cos(eps)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) + math.tan(x)
	t_1 = math.tan(eps) * math.tan(x)
	t_2 = 1.0 - t_1
	tmp = 0
	if eps <= -5.8e-9:
		tmp = (t_0 / t_2) - math.tan(x)
	elif eps <= 6.2e-9:
		tmp = eps + (math.tan(x) * (t_1 / t_2))
	else:
		tmp = (t_0 / (1.0 - ((math.tan(x) * math.sin(eps)) / math.cos(eps)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(tan(eps) * tan(x))
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (eps <= -5.8e-9)
		tmp = Float64(Float64(t_0 / t_2) - tan(x));
	elseif (eps <= 6.2e-9)
		tmp = Float64(eps + Float64(tan(x) * Float64(t_1 / t_2)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) + tan(x);
	t_1 = tan(eps) * tan(x);
	t_2 = 1.0 - t_1;
	tmp = 0.0;
	if (eps <= -5.8e-9)
		tmp = (t_0 / t_2) - tan(x);
	elseif (eps <= 6.2e-9)
		tmp = eps + (tan(x) * (t_1 / t_2));
	else
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[eps, -5.8e-9], N[(N[(t$95$0 / t$95$2), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6.2e-9], N[(eps + N[(N[Tan[x], $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.79999999999999982e-9

   1. Initial program 65.7%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.6%

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

      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 99.7%

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

   if -5.79999999999999982e-9 < eps < 6.2000000000000001e-9

   1. Initial program 35.3%

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

      1. tan-sum 35.3%

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

      2. div-inv 35.3%

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

      3. *-un-lft-identity 35.3%

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

      4. prod-diff 35.3%

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

      5. *-commutative 35.3%

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

      6. *-un-lft-identity 35.3%

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

      7. *-commutative 35.3%

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

      8. *-un-lft-identity 35.3%

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

   4. Applied egg-rr 35.3%

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

      1. +-commutative 35.3%

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

      2. fma-udef 35.3%

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

      3. associate-+r+ 35.3%

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

      4. unsub-neg 35.3%

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

   6. Simplified 35.3%

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

   7. Taylor expanded in x around inf 35.3%

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

   9. Simplified 61.9%

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

       1. tan-quot 58.1%

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

       2. clear-num 55.3%

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

       3. frac-sub 51.5%

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

   11. Applied egg-rr 53.3%

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

       1. associate-/r/ 53.3%

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

       2. rgt-mult-inverse 61.9%

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

       3. associate--r- 99.8%

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

       4. metadata-eval 99.8%

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

       5. +-lft-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in eps around 0 99.8%

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

   if 6.2000000000000001e-9 < eps

   1. Initial program 56.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.2%

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

      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 99.3%

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

      1. tan-quot 99.4%

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

      2. associate-*r/ 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \tan x \cdot \frac{\tan \varepsilon \cdot \tan x}{1 - \tan \varepsilon \cdot \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon \cdot \tan x\\ t_1 := 1 - t_0\\ \frac{\tan \varepsilon}{t_1} + \tan x \cdot \frac{t_0}{t_1} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan eps) (tan x))) (t_1 (- 1.0 t_0)))
   (+ (/ (tan eps) t_1) (* (tan x) (/ t_0 t_1)))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) * tan(x);
	double t_1 = 1.0 - t_0;
	return (tan(eps) / t_1) + (tan(x) * (t_0 / t_1));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = tan(eps) * tan(x)
    t_1 = 1.0d0 - t_0
    code = (tan(eps) / t_1) + (tan(x) * (t_0 / t_1))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) * Math.tan(x);
	double t_1 = 1.0 - t_0;
	return (Math.tan(eps) / t_1) + (Math.tan(x) * (t_0 / t_1));
}

Python

def code(x, eps):
	t_0 = math.tan(eps) * math.tan(x)
	t_1 = 1.0 - t_0
	return (math.tan(eps) / t_1) + (math.tan(x) * (t_0 / t_1))

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) * tan(x))
	t_1 = Float64(1.0 - t_0)
	return Float64(Float64(tan(eps) / t_1) + Float64(tan(x) * Float64(t_0 / t_1)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(eps) * tan(x);
	t_1 = 1.0 - t_0;
	tmp = (tan(eps) / t_1) + (tan(x) * (t_0 / t_1));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, N[(N[(N[Tan[eps], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 48.4%

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

   1. tan-sum 67.9%

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

   2. div-inv 67.8%

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

   3. *-un-lft-identity 67.8%

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

   4. prod-diff 67.8%

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

   5. *-commutative 67.8%

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

   6. *-un-lft-identity 67.8%

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

   7. *-commutative 67.8%

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

   8. *-un-lft-identity 67.8%

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

4. Applied egg-rr 67.8%

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

   1. +-commutative 67.8%

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

   2. fma-udef 67.8%

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

   3. associate-+r+ 67.8%

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

   4. unsub-neg 67.8%

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

6. Simplified 67.9%

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

7. Taylor expanded in x around inf 67.7%

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

9. Simplified 80.8%

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

    1. tan-quot 78.9%

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

    2. clear-num 77.5%

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

    3. frac-sub 75.6%

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

11. Applied egg-rr 76.5%

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

    1. associate-/r/ 76.5%

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

    2. rgt-mult-inverse 80.8%

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

    3. associate--r- 99.5%

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

    4. metadata-eval 99.5%

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

    5. +-lft-identity 99.5%

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

13. Simplified 99.5%

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

    1. expm1-log1p-u 92.4%

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

    2. expm1-udef 49.1%

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

    3. associate-/r* 49.1%

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

    4. tan-quot 49.2%

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

    5. tan-quot 49.2%

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

    6. tan-quot 49.2%

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

    7. *-commutative 49.2%

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

15. Applied egg-rr 49.2%

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

    1. expm1-def 92.5%

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

    2. expm1-log1p 99.7%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon \cdot \tan x\\ t_1 := 1 - t_0\\ \mathbf{if}\;\varepsilon \leq -7.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_1} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \tan x \cdot \frac{t_0}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan eps) (tan x))) (t_1 (- 1.0 t_0)))
   (if (or (<= eps -7.3e-9) (not (<= eps 5.8e-9)))
     (- (/ (+ (tan eps) (tan x)) t_1) (tan x))
     (+ eps (* (tan x) (/ t_0 t_1))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) * tan(x);
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((eps <= -7.3e-9) || !(eps <= 5.8e-9)) {
		tmp = ((tan(eps) + tan(x)) / t_1) - tan(x);
	} else {
		tmp = eps + (tan(x) * (t_0 / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(eps) * tan(x)
    t_1 = 1.0d0 - t_0
    if ((eps <= (-7.3d-9)) .or. (.not. (eps <= 5.8d-9))) then
        tmp = ((tan(eps) + tan(x)) / t_1) - tan(x)
    else
        tmp = eps + (tan(x) * (t_0 / t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) * Math.tan(x);
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((eps <= -7.3e-9) || !(eps <= 5.8e-9)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / t_1) - Math.tan(x);
	} else {
		tmp = eps + (Math.tan(x) * (t_0 / t_1));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) * math.tan(x)
	t_1 = 1.0 - t_0
	tmp = 0
	if (eps <= -7.3e-9) or not (eps <= 5.8e-9):
		tmp = ((math.tan(eps) + math.tan(x)) / t_1) - math.tan(x)
	else:
		tmp = eps + (math.tan(x) * (t_0 / t_1))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) * tan(x))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if ((eps <= -7.3e-9) || !(eps <= 5.8e-9))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / t_1) - tan(x));
	else
		tmp = Float64(eps + Float64(tan(x) * Float64(t_0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) * tan(x);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if ((eps <= -7.3e-9) || ~((eps <= 5.8e-9)))
		tmp = ((tan(eps) + tan(x)) / t_1) - tan(x);
	else
		tmp = eps + (tan(x) * (t_0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[Or[LessEqual[eps, -7.3e-9], N[Not[LessEqual[eps, 5.8e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(N[Tan[x], $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.30000000000000002e-9 or 5.79999999999999982e-9 < eps

   1. Initial program 61.1%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   6. Simplified 99.5%

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

   if -7.30000000000000002e-9 < eps < 5.79999999999999982e-9

   1. Initial program 35.3%

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

      1. tan-sum 35.3%

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

      2. div-inv 35.3%

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

      3. *-un-lft-identity 35.3%

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

      4. prod-diff 35.3%

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

      5. *-commutative 35.3%

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

      6. *-un-lft-identity 35.3%

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

      7. *-commutative 35.3%

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

      8. *-un-lft-identity 35.3%

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

   4. Applied egg-rr 35.3%

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

      1. +-commutative 35.3%

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

      2. fma-udef 35.3%

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

      3. associate-+r+ 35.3%

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

      4. unsub-neg 35.3%

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

   6. Simplified 35.3%

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

   7. Taylor expanded in x around inf 35.3%

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

   9. Simplified 61.9%

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

       1. tan-quot 58.1%

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

       2. clear-num 55.3%

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

       3. frac-sub 51.5%

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

   11. Applied egg-rr 53.3%

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

       1. associate-/r/ 53.3%

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

       2. rgt-mult-inverse 61.9%

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

       3. associate--r- 99.8%

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

       4. metadata-eval 99.8%

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

       5. +-lft-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \tan x \cdot \frac{\tan \varepsilon \cdot \tan x}{1 - \tan \varepsilon \cdot \tan x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.6e-9) (not (<= eps 3.3e-9)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (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 <= -3.6e-9) || !(eps <= 3.3e-9)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - 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 <= (-3.6d-9)) .or. (.not. (eps <= 3.3d-9))) then
        tmp = ((tan(eps) + tan(x)) / (1.0d0 - (tan(eps) * tan(x)))) - 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 <= -3.6e-9) || !(eps <= 3.3e-9)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / (1.0 - (Math.tan(eps) * Math.tan(x)))) - 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 <= -3.6e-9) or not (eps <= 3.3e-9):
		tmp = ((math.tan(eps) + math.tan(x)) / (1.0 - (math.tan(eps) * math.tan(x)))) - 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 <= -3.6e-9) || !(eps <= 3.3e-9))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - 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 <= -3.6e-9) || ~((eps <= 3.3e-9)))
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - 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, -3.6e-9], N[Not[LessEqual[eps, 3.3e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(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 < -3.6e-9 or 3.30000000000000018e-9 < eps

   1. Initial program 61.1%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   6. Simplified 99.5%

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

   if -3.6e-9 < eps < 3.30000000000000018e-9

   1. Initial program 35.3%

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

   3. Taylor expanded in eps around 0 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.13:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5e-6)
   (- (log1p (expm1 (/ (sin eps) (cos eps)))) (tan x))
   (if (<= eps 0.13)
     (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5e-6) {
		tmp = log1p(expm1((sin(eps) / cos(eps)))) - tan(x);
	} else if (eps <= 0.13) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -5e-6) {
		tmp = Math.log1p(Math.expm1((Math.sin(eps) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 0.13) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -5e-6:
		tmp = math.log1p(math.expm1((math.sin(eps) / math.cos(eps)))) - math.tan(x)
	elif eps <= 0.13:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5e-6)
		tmp = Float64(log1p(expm1(Float64(sin(eps) / cos(eps)))) - tan(x));
	elseif (eps <= 0.13)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5e-6], N[(N[Log[1 + N[(Exp[N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.13], 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], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.00000000000000041e-6

   1. Initial program 65.7%

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

      1. log1p-expm1-u 65.7%

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

   4. Applied egg-rr 65.7%

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

   5. Taylor expanded in x around 0 68.0%

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

   7. Simplified 68.5%

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

   if -5.00000000000000041e-6 < eps < 0.13

   1. Initial program 35.5%

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

   3. Taylor expanded in eps around 0 98.7%

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

      1. cancel-sign-sub-inv 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

   5. Simplified 98.7%

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

   if 0.13 < eps

   1. Initial program 56.4%

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

   3. Taylor expanded in x around 0 59.3%

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

      1. tan-quot 59.6%

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

      2. expm1-log1p-u 47.8%

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

      3. expm1-udef 47.6%

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

   5. Applied egg-rr 47.6%

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

   7. Simplified 59.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.13:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.13\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7e-6) (not (<= eps 0.13)))
   (tan eps)
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-6) || !(eps <= 0.13)) {
		tmp = tan(eps);
	} 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 <= (-1.7d-6)) .or. (.not. (eps <= 0.13d0))) then
        tmp = tan(eps)
    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 <= -1.7e-6) || !(eps <= 0.13)) {
		tmp = Math.tan(eps);
	} 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 <= -1.7e-6) or not (eps <= 0.13):
		tmp = math.tan(eps)
	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 <= -1.7e-6) || !(eps <= 0.13))
		tmp = tan(eps);
	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 <= -1.7e-6) || ~((eps <= 0.13)))
		tmp = tan(eps);
	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, -1.7e-6], N[Not[LessEqual[eps, 0.13]], $MachinePrecision]], N[Tan[eps], $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 < -1.70000000000000003e-6 or 0.13 < eps

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0 63.7%

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

      1. tan-quot 64.0%

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

      2. expm1-log1p-u 50.5%

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

      3. expm1-udef 50.3%

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

   5. Applied egg-rr 50.3%

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

   7. Simplified 64.0%

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

   if -1.70000000000000003e-6 < eps < 0.13

   1. Initial program 35.5%

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

   3. Taylor expanded in eps around 0 98.7%

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

      1. cancel-sign-sub-inv 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.13\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.4%

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

3. Taylor expanded in x around 0 62.5%

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

   1. tan-quot 62.7%

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

   2. expm1-log1p-u 55.9%

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

   3. expm1-udef 28.1%

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

5. Applied egg-rr 28.1%

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

7. Simplified 62.7%

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

8. Final simplification 62.7%

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

Alternative 8: 30.8% 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 48.4%

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

3. Taylor expanded in x around 0 62.5%

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

4. Taylor expanded in eps around 0 32.5%

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

5. Final simplification 32.5%

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

Developer target: 76.0% 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 2024014 
(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)))