2tan (problem 3.3.2)

Percentage Accurate: 42.2% → 99.5%

Time: 20.8s

Alternatives: 10

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: 42.2% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ t_2 := {\sin x}^{2}\\ t_3 := {\cos x}^{2}\\ t_4 := \tan x + \tan \varepsilon\\ t_5 := \frac{t_2}{t_3}\\ t_6 := t_5 + 1\\ \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_4, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0\right) + t_1\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_6, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(\frac{t_2}{\frac{t_3}{t_6}} - t_5 \cdot -0.3333333333333333\right), \frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot t_6}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \mathsf{fma}\left(t_4, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x)))
        (t_1 (fma -1.0 (tan x) (tan x)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (pow (cos x) 2.0))
        (t_4 (+ (tan x) (tan eps)))
        (t_5 (/ t_2 t_3))
        (t_6 (+ t_5 1.0)))
   (if (<= eps -6.6e-5)
     (+ (fma t_4 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_0) t_1)
     (if (<= eps 4.5e-5)
       (fma
        eps
        t_6
        (fma
         (pow eps 3.0)
         (+
          0.3333333333333333
          (- (/ t_2 (/ t_3 t_6)) (* t_5 -0.3333333333333333)))
         (/ (pow eps 2.0) (/ (cos x) (* (sin x) t_6)))))
       (+ t_1 (fma t_4 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_0))))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = fma(-1.0, tan(x), tan(x));
	double t_2 = pow(sin(x), 2.0);
	double t_3 = pow(cos(x), 2.0);
	double t_4 = tan(x) + tan(eps);
	double t_5 = t_2 / t_3;
	double t_6 = t_5 + 1.0;
	double tmp;
	if (eps <= -6.6e-5) {
		tmp = fma(t_4, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_0) + t_1;
	} else if (eps <= 4.5e-5) {
		tmp = fma(eps, t_6, fma(pow(eps, 3.0), (0.3333333333333333 + ((t_2 / (t_3 / t_6)) - (t_5 * -0.3333333333333333))), (pow(eps, 2.0) / (cos(x) / (sin(x) * t_6)))));
	} else {
		tmp = t_1 + fma(t_4, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = fma(-1.0, tan(x), tan(x))
	t_2 = sin(x) ^ 2.0
	t_3 = cos(x) ^ 2.0
	t_4 = Float64(tan(x) + tan(eps))
	t_5 = Float64(t_2 / t_3)
	t_6 = Float64(t_5 + 1.0)
	tmp = 0.0
	if (eps <= -6.6e-5)
		tmp = Float64(fma(t_4, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_0) + t_1);
	elseif (eps <= 4.5e-5)
		tmp = fma(eps, t_6, fma((eps ^ 3.0), Float64(0.3333333333333333 + Float64(Float64(t_2 / Float64(t_3 / t_6)) - Float64(t_5 * -0.3333333333333333))), Float64((eps ^ 2.0) / Float64(cos(x) / Float64(sin(x) * t_6)))));
	else
		tmp = Float64(t_1 + fma(t_4, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + 1.0), $MachinePrecision]}, If[LessEqual[eps, -6.6e-5], N[(N[(t$95$4 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 4.5e-5], N[(eps * t$95$6 + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(t$95$2 / N[(t$95$3 / t$95$6), $MachinePrecision]), $MachinePrecision] - N[(t$95$5 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$4 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.6000000000000005e-5

   1. Initial program 58.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. *-commutative 99.5%

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

      5. prod-diff 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-un-lft-identity 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -6.6000000000000005e-5 < eps < 4.50000000000000028e-5

   1. Initial program 25.0%

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

      1. tan-sum 26.2%

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

      2. div-inv 26.1%

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

      3. *-un-lft-identity 26.1%

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

      4. prod-diff 26.1%

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

      5. *-commutative 26.1%

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

      6. *-un-lft-identity 26.1%

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

      7. *-commutative 26.1%

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

      8. *-un-lft-identity 26.1%

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

   3. Applied egg-rr 26.1%

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

   5. Simplified 26.2%

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

      1. add-cube-cbrt 24.6%

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

      2. pow3 24.6%

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

   7. Applied egg-rr 24.6%

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

   8. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]
   9. Step-by-step derivation

      1. fma-def 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

   if 4.50000000000000028e-5 < eps

   1. Initial program 43.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. 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. *-commutative 99.2%

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

      5. prod-diff 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. /-rgt-identity 99.3%

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

      2. frac-2neg 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-def 99.5%

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

   7. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(\frac{{\sin x}^{2}}{\frac{{\cos x}^{2}}{\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1}} - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333\right), \frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ t_2 := \tan x + \tan \varepsilon\\ t_3 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0\right) + t_1\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t_3 - \left(t_3 \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) + \varepsilon \cdot \left(t_3 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \mathsf{fma}\left(t_2, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x)))
        (t_1 (fma -1.0 (tan x) (tan x)))
        (t_2 (+ (tan x) (tan eps)))
        (t_3 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (if (<= eps -5.2e-5)
     (+ (fma t_2 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_0) t_1)
     (if (<= eps 5.9e-5)
       (+
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0))))
        (+
         (*
          (pow eps 3.0)
          (+
           0.3333333333333333
           (-
            t_3
            (-
             (* t_3 -0.3333333333333333)
             (/ (pow (sin x) 4.0) (pow (cos x) 4.0))))))
         (* eps (+ t_3 1.0))))
       (+ t_1 (fma t_2 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_0))))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = fma(-1.0, tan(x), tan(x));
	double t_2 = tan(x) + tan(eps);
	double t_3 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double tmp;
	if (eps <= -5.2e-5) {
		tmp = fma(t_2, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_0) + t_1;
	} else if (eps <= 5.9e-5) {
		tmp = (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))) + ((pow(eps, 3.0) * (0.3333333333333333 + (t_3 - ((t_3 * -0.3333333333333333) - (pow(sin(x), 4.0) / pow(cos(x), 4.0)))))) + (eps * (t_3 + 1.0)));
	} else {
		tmp = t_1 + fma(t_2, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = fma(-1.0, tan(x), tan(x))
	t_2 = Float64(tan(x) + tan(eps))
	t_3 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	tmp = 0.0
	if (eps <= -5.2e-5)
		tmp = Float64(fma(t_2, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_0) + t_1);
	elseif (eps <= 5.9e-5)
		tmp = Float64(Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))) + Float64(Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(t_3 - Float64(Float64(t_3 * -0.3333333333333333) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))))) + Float64(eps * Float64(t_3 + 1.0))));
	else
		tmp = Float64(t_1 + fma(t_2, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.2e-5], N[(N[(t$95$2 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 5.9e-5], N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$3 - N[(N[(t$95$3 * -0.3333333333333333), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$2 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.19999999999999968e-5

   1. Initial program 58.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. *-commutative 99.5%

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

      5. prod-diff 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-un-lft-identity 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -5.19999999999999968e-5 < eps < 5.8999999999999998e-5

   1. Initial program 25.0%

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

      1. tan-sum 26.2%

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

      2. div-inv 26.1%

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

      3. *-un-lft-identity 26.1%

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

      4. prod-diff 26.1%

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

      5. *-commutative 26.1%

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

      6. *-un-lft-identity 26.1%

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

      7. *-commutative 26.1%

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

      8. *-un-lft-identity 26.1%

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

   3. Applied egg-rr 26.1%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in eps around 0 99.5%

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

   if 5.8999999999999998e-5 < eps

   1. Initial program 43.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. 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. *-commutative 99.2%

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

      5. prod-diff 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. /-rgt-identity 99.3%

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

      2. frac-2neg 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-def 99.5%

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

   7. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) + \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\\ \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0 - \tan \varepsilon}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) - \varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{t_1}, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (fma (tan x) (tan eps) -1.0)))
   (if (<= eps -8e-7)
     (- (/ (- t_0 (tan eps)) t_1) (tan x))
     (if (<= eps 8.2e-7)
       (-
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0))))
        (* eps (- -1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
       (+
        (fma -1.0 (tan x) (tan x))
        (fma (+ (tan x) (tan eps)) (/ -1.0 t_1) t_0))))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = fma(tan(x), tan(eps), -1.0);
	double tmp;
	if (eps <= -8e-7) {
		tmp = ((t_0 - tan(eps)) / t_1) - tan(x);
	} else if (eps <= 8.2e-7) {
		tmp = (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))) - (eps * (-1.0 - (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
	} else {
		tmp = fma(-1.0, tan(x), tan(x)) + fma((tan(x) + tan(eps)), (-1.0 / t_1), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = fma(tan(x), tan(eps), -1.0)
	tmp = 0.0
	if (eps <= -8e-7)
		tmp = Float64(Float64(Float64(t_0 - tan(eps)) / t_1) - tan(x));
	elseif (eps <= 8.2e-7)
		tmp = Float64(Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))) - Float64(eps * Float64(-1.0 - Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))));
	else
		tmp = Float64(fma(-1.0, tan(x), tan(x)) + fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / t_1), t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -8e-7], N[(N[(N[(t$95$0 - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 8.2e-7], N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision], N[(N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.9999999999999996e-7

   1. Initial program 58.1%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-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 99.3%

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

      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%

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

   3. Applied egg-rr 99.3%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 42.5%

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

      8. sqrt-unprod 81.6%

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

      9. sqr-neg 81.6%

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

      10. sqrt-unprod 39.0%

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

      11. add-sqr-sqrt 60.6%

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

      12. rem-cbrt-cube 60.6%

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

      13. add-sqr-sqrt 54.9%

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

      14. add-sqr-sqrt 60.6%

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

      15. rem-cbrt-cube 60.6%

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

      16. distribute-lft-neg-in 60.6%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   9. Simplified 99.4%

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

   if -7.9999999999999996e-7 < eps < 8.1999999999999998e-7

   1. Initial program 24.6%

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

      1. tan-sum 25.2%

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

      2. div-inv 25.1%

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

      3. *-un-lft-identity 25.1%

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

      4. prod-diff 25.1%

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

      5. *-commutative 25.1%

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

      6. *-un-lft-identity 25.1%

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

      7. *-commutative 25.1%

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

      8. *-un-lft-identity 25.1%

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

   3. Applied egg-rr 25.1%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in eps around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. cancel-sign-sub-inv 99.4%

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

      5. metadata-eval 99.4%

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

      6. *-lft-identity 99.4%

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

   8. Simplified 99.4%

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

   if 8.1999999999999998e-7 < eps

   1. Initial program 43.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. 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. *-commutative 99.2%

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

      5. prod-diff 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. /-rgt-identity 99.3%

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

      2. frac-2neg 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-def 99.5%

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

   7. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) - \varepsilon \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0 - \tan \varepsilon}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{t_1}, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (fma (tan x) (tan eps) -1.0)))
   (if (<= eps -3.6e-9)
     (- (/ (- t_0 (tan eps)) t_1) (tan x))
     (if (<= eps 5.1e-9)
       (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
       (+
        (fma -1.0 (tan x) (tan x))
        (fma (+ (tan x) (tan eps)) (/ -1.0 t_1) t_0))))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = fma(tan(x), tan(eps), -1.0);
	double tmp;
	if (eps <= -3.6e-9) {
		tmp = ((t_0 - tan(eps)) / t_1) - tan(x);
	} else if (eps <= 5.1e-9) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = fma(-1.0, tan(x), tan(x)) + fma((tan(x) + tan(eps)), (-1.0 / t_1), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = fma(tan(x), tan(eps), -1.0)
	tmp = 0.0
	if (eps <= -3.6e-9)
		tmp = Float64(Float64(Float64(t_0 - tan(eps)) / t_1) - tan(x));
	elseif (eps <= 5.1e-9)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(fma(-1.0, tan(x), tan(x)) + fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / t_1), t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -3.6e-9], N[(N[(N[(t$95$0 - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.1e-9], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.6e-9

   1. Initial program 58.1%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-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 99.3%

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

      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%

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

   3. Applied egg-rr 99.3%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 42.5%

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

      8. sqrt-unprod 81.6%

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

      9. sqr-neg 81.6%

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

      10. sqrt-unprod 39.0%

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

      11. add-sqr-sqrt 60.6%

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

      12. rem-cbrt-cube 60.6%

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

      13. add-sqr-sqrt 54.9%

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

      14. add-sqr-sqrt 60.6%

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

      15. rem-cbrt-cube 60.6%

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

      16. distribute-lft-neg-in 60.6%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   9. Simplified 99.4%

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

   if -3.6e-9 < eps < 5.10000000000000017e-9

   1. Initial program 24.8%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   4. Simplified 99.5%

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

   if 5.10000000000000017e-9 < eps

   1. Initial program 42.7%

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

      1. tan-sum 98.7%

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

      2. div-inv 98.7%

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

      3. *-un-lft-identity 98.7%

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

      4. *-commutative 98.7%

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

      5. prod-diff 98.8%

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

      6. *-un-lft-identity 98.8%

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

      7. metadata-eval 98.8%

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

      8. *-un-lft-identity 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. /-rgt-identity 98.8%

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

      2. frac-2neg 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. fma-def 99.0%

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

   7. Simplified 99.0%

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

      1. clear-num 99.0%

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

      2. div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.0%

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

       2. metadata-eval 99.0%

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

   11. Simplified 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.75e-9)
   (- (/ (- (- (tan x)) (tan eps)) (fma (tan x) (tan eps) -1.0)) (tan x))
   (if (<= eps 4.7e-9)
     (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan eps) (/ (cos x) (sin x)))))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.75e-9) {
		tmp = ((-tan(x) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
	} else if (eps <= 4.7e-9) {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	} else {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2.75e-9)
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x));
	elseif (eps <= 4.7e-9)
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.75e-9], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.7e-9], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.7499999999999998e-9

   1. Initial program 58.1%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-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 99.3%

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

      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%

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

   3. Applied egg-rr 99.3%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. add-sqr-sqrt 42.5%

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

      8. sqrt-unprod 81.6%

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

      9. sqr-neg 81.6%

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

      10. sqrt-unprod 39.0%

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

      11. add-sqr-sqrt 60.6%

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

      12. rem-cbrt-cube 60.6%

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

      13. add-sqr-sqrt 54.9%

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

      14. add-sqr-sqrt 60.6%

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

      15. rem-cbrt-cube 60.6%

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

      16. distribute-lft-neg-in 60.6%

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

   7. Applied egg-rr 99.4%

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

      1. distribute-neg-frac 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-def 99.4%

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

   9. Simplified 99.4%

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

   if -2.7499999999999998e-9 < eps < 4.6999999999999999e-9

   1. Initial program 24.8%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   4. Simplified 99.5%

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

   if 4.6999999999999999e-9 < eps

   1. Initial program 42.7%

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

      1. tan-sum 98.7%

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

      2. div-inv 98.7%

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

      3. *-un-lft-identity 98.7%

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

      4. prod-diff 98.8%

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

      5. *-commutative 98.8%

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

      6. *-un-lft-identity 98.8%

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

      7. *-commutative 98.8%

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

      8. *-un-lft-identity 98.8%

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

   3. Applied egg-rr 98.8%

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

   5. Simplified 98.7%

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

      1. *-commutative 98.7%

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

      2. tan-quot 98.8%

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

      3. associate-*r/ 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. associate-/l* 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5e-9) (not (<= eps 3.4e-9)))
   (- (/ (- (- (tan x)) (tan eps)) (fma (tan x) (tan eps) -1.0)) (tan x))
   (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5e-9) || !(eps <= 3.4e-9)) {
		tmp = ((-tan(x) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
	} else {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5e-9) || !(eps <= 3.4e-9))
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x));
	else
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5e-9], N[Not[LessEqual[eps, 3.4e-9]], $MachinePrecision]], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.0000000000000001e-9 or 3.3999999999999998e-9 < eps

   1. Initial program 50.5%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

   5. Simplified 99.1%

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

      1. frac-2neg 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. distribute-neg-in 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-lft-neg-in 99.1%

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

      7. add-sqr-sqrt 43.2%

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

      8. sqrt-unprod 77.4%

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

      9. sqr-neg 77.4%

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

      10. sqrt-unprod 34.1%

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

      11. add-sqr-sqrt 54.2%

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

      12. rem-cbrt-cube 54.2%

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

      13. add-sqr-sqrt 47.8%

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

      14. add-sqr-sqrt 54.2%

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

      15. rem-cbrt-cube 54.2%

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

      16. distribute-lft-neg-in 54.2%

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

   7. Applied egg-rr 99.1%

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

      1. distribute-neg-frac 99.1%

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

      2. +-commutative 99.1%

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

      3. fma-def 99.1%

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

   9. Simplified 99.1%

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

   if -5.0000000000000001e-9 < eps < 3.3999999999999998e-9

   1. Initial program 24.8%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \]

Alternative 7: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.8e-9) (not (<= eps 5e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-9) || !(eps <= 5e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.8d-9)) .or. (.not. (eps <= 5d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-9) || !(eps <= 5e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.8e-9) or not (eps <= 5e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.8e-9) || !(eps <= 5e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.8e-9) || ~((eps <= 5e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.8e-9], N[Not[LessEqual[eps, 5e-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[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.80000000000000011e-9 or 5.0000000000000001e-9 < eps

   1. Initial program 50.5%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

   5. Simplified 99.1%

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

   if -3.80000000000000011e-9 < eps < 5.0000000000000001e-9

   1. Initial program 24.8%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \]

Alternative 8: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0285 \lor \neg \left(\varepsilon \leq 0.00068\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0285) (not (<= eps 0.00068)))
   (tan eps)
   (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0285) || !(eps <= 0.00068)) {
		tmp = tan(eps);
	} else {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0285d0)) .or. (.not. (eps <= 0.00068d0))) then
        tmp = tan(eps)
    else
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0285) || !(eps <= 0.00068)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0285) or not (eps <= 0.00068):
		tmp = math.tan(eps)
	else:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0285) || !(eps <= 0.00068))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0285) || ~((eps <= 0.00068)))
		tmp = tan(eps);
	else
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0285], N[Not[LessEqual[eps, 0.00068]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.028500000000000001 or 6.8e-4 < eps

   1. Initial program 51.3%

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

   2. Taylor expanded in x around 0 53.9%

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

      1. tan-quot 54.1%

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

      2. add-log-exp 53.0%

         \[\leadsto \color{blue}{\log \left(e^{\tan \varepsilon}\right)} \]

      3. *-un-lft-identity 53.0%

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

      4. log-prod 53.0%

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

      5. metadata-eval 53.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\tan \varepsilon}\right) \]

      6. add-log-exp 54.1%

         \[\leadsto 0 + \color{blue}{\tan \varepsilon} \]

   4. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{0 + \tan \varepsilon} \]
   5. Step-by-step derivation

      1. +-lft-identity 54.1%

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

   6. Simplified 54.1%

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

   if -0.028500000000000001 < eps < 6.8e-4

   1. Initial program 24.9%

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

   2. Taylor expanded in eps around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. *-lft-identity 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0285 \lor \neg \left(\varepsilon \leq 0.00068\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \]

Alternative 9: 58.4% 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 37.3%

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

2. Taylor expanded in x around 0 55.8%

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

   1. tan-quot 55.8%

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

   2. add-log-exp 28.4%

      \[\leadsto \color{blue}{\log \left(e^{\tan \varepsilon}\right)} \]

   3. *-un-lft-identity 28.4%

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

   4. log-prod 28.4%

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

   5. metadata-eval 28.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\tan \varepsilon}\right) \]

   6. add-log-exp 55.8%

      \[\leadsto 0 + \color{blue}{\tan \varepsilon} \]

4. Applied egg-rr 55.8%

   \[\leadsto \color{blue}{0 + \tan \varepsilon} \]
5. Step-by-step derivation

   1. +-lft-identity 55.8%

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

6. Simplified 55.8%

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

7. Final simplification 55.8%

   \[\leadsto \tan \varepsilon \]

Alternative 10: 31.3% 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 37.3%

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

2. Taylor expanded in x around 0 55.8%

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

3. Taylor expanded in eps around 0 31.9%

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

4. Final simplification 31.9%

   \[\leadsto \varepsilon \]

Developer target: 76.8% 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 2023335 
(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)))