2tan (problem 3.3.2)

Average Accuracy: 42.1% → 98.9%

Time: 24.6s

Precision: binary64

Cost: 131016

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{\sin x}{\cos x}\\ t_3 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -1.28 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}, t_3\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot t_2} - \tan x \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \frac{-0.3333333333333333}{\cos x} - {t_2}^{3}\right) - \mathsf{fma}\left(\frac{\varepsilon}{\cos x}, \sin x, {\sin x}^{2} \cdot \frac{\varepsilon \cdot \varepsilon}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_3\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (/ (sin x) (cos x)))
        (t_3 (- (tan x))))
   (if (<= eps -1.28e+16)
     (fma t_1 (/ 1.0 (fma (tan x) (- (tan eps)) 1.0)) t_3)
     (if (<= eps 2.1e-11)
       (-
        (/ t_0 (- 1.0 (* t_0 t_2)))
        (*
         (tan x)
         (-
          (*
           (pow eps 3.0)
           (- (* (sin x) (/ -0.3333333333333333 (cos x))) (pow t_2 3.0)))
          (fma
           (/ eps (cos x))
           (sin x)
           (* (pow (sin x) 2.0) (/ (* eps eps) (pow (cos x) 2.0)))))))
       (fma t_1 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_3)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double t_1 = tan(x) + tan(eps);
	double t_2 = sin(x) / cos(x);
	double t_3 = -tan(x);
	double tmp;
	if (eps <= -1.28e+16) {
		tmp = fma(t_1, (1.0 / fma(tan(x), -tan(eps), 1.0)), t_3);
	} else if (eps <= 2.1e-11) {
		tmp = (t_0 / (1.0 - (t_0 * t_2))) - (tan(x) * ((pow(eps, 3.0) * ((sin(x) * (-0.3333333333333333 / cos(x))) - pow(t_2, 3.0))) - fma((eps / cos(x)), sin(x), (pow(sin(x), 2.0) * ((eps * eps) / pow(cos(x), 2.0))))));
	} else {
		tmp = fma(t_1, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_3);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	t_1 = Float64(tan(x) + tan(eps))
	t_2 = Float64(sin(x) / cos(x))
	t_3 = Float64(-tan(x))
	tmp = 0.0
	if (eps <= -1.28e+16)
		tmp = fma(t_1, Float64(1.0 / fma(tan(x), Float64(-tan(eps)), 1.0)), t_3);
	elseif (eps <= 2.1e-11)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * t_2))) - Float64(tan(x) * Float64(Float64((eps ^ 3.0) * Float64(Float64(sin(x) * Float64(-0.3333333333333333 / cos(x))) - (t_2 ^ 3.0))) - fma(Float64(eps / cos(x)), sin(x), Float64((sin(x) ^ 2.0) * Float64(Float64(eps * eps) / (cos(x) ^ 2.0)))))));
	else
		tmp = fma(t_1, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_3);
	end
	return tmp
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Tan[x], $MachinePrecision])}, If[LessEqual[eps, -1.28e+16], N[(t$95$1 * N[(1.0 / N[(N[Tan[x], $MachinePrecision] * (-N[Tan[eps], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[eps, 2.1e-11], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Tan[x], $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[(-0.3333333333333333 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]

Target

Original	42.1%
Target	76.2%
Herbie	98.9%

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.28e16

   1. Initial program 54.3%

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

   2. Applied egg-rr 99.5%

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

      [Start] 54.3	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.4	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.5	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 99.5%

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

      [Start] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      sub-neg [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{1 + \left(-\tan x \cdot \tan \varepsilon\right)}}, -\tan x\right) \]
      +-commutative [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\left(-\tan x \cdot \tan \varepsilon\right) + 1}}, -\tan x\right) \]
      distribute-rgt-neg-in [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\tan x \cdot \left(-\tan \varepsilon\right)} + 1}, -\tan x\right) \]
      fma-def [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}}, -\tan x\right) \]

   if -1.28e16 < eps < 2.0999999999999999e-11

   1. Initial program 30.6%

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

   2. Applied egg-rr 32.5%

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

      [Start] 30.6	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 32.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 32.5	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 32.5	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Taylor expanded in x around inf 32.5%

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

   4. Simplified 61.3%

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

      [Start] 32.5	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
      associate--l+ [=>] 61.3	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]
      associate-/r* [=>] 61.3	\[ \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]
      *-commutative [<=] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\color{blue}{\cos x \cdot \cos \varepsilon}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]
      times-frac [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]
      *-commutative [<=] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]

   5. Applied egg-rr 61.3%

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

      [Start] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) \]
      frac-2neg [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\color{blue}{\frac{-\frac{\sin x}{\cos x}}{-\left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)}} - \frac{\sin x}{\cos x}\right) \]
      div-inv [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\color{blue}{\left(-\frac{\sin x}{\cos x}\right) \cdot \frac{1}{-\left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)}} - \frac{\sin x}{\cos x}\right) \]
      fma-neg [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \color{blue}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{1}{-\left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)}, -\frac{\sin x}{\cos x}\right)} \]

   6. Simplified 61.3%

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

      [Start] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \mathsf{fma}\left(-\tan x, \frac{1}{-\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}, -\tan x\right) \]
      fma-udef [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \color{blue}{\left(\left(-\tan x\right) \cdot \frac{1}{-\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} + \left(-\tan x\right)\right)} \]
      *-rgt-identity [<=] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\left(-\tan x\right) \cdot \frac{1}{-\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} + \color{blue}{\left(-\tan x\right) \cdot 1}\right) \]
      distribute-lft-out [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \color{blue}{\left(-\tan x\right) \cdot \left(\frac{1}{-\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} + 1\right)} \]
      neg-mul-1 [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(\frac{1}{\color{blue}{-1 \cdot \mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}} + 1\right) \]
      associate-/r* [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(\color{blue}{\frac{\frac{1}{-1}}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}} + 1\right) \]
      metadata-eval [=>] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(\frac{\color{blue}{-1}}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} + 1\right) \]

   7. Taylor expanded in eps around 0 98.4%

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

   8. Simplified 98.4%

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

      [Start] 98.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(-1 \cdot \frac{{\varepsilon}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) + -1 \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\right)\right) \]
      +-commutative [=>] 98.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \color{blue}{\left(\left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) + -1 \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\right) + -1 \cdot \frac{{\varepsilon}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)} \]
      mul-1-neg [=>] 98.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(\left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) + -1 \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\right) + \color{blue}{\left(-\frac{{\varepsilon}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)}\right) \]
      unsub-neg [=>] 98.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \color{blue}{\left(\left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) + -1 \cdot \frac{\varepsilon \cdot \sin x}{\cos x}\right) - \frac{{\varepsilon}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)} \]
      mul-1-neg [=>] 98.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(\left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) + \color{blue}{\left(-\frac{\varepsilon \cdot \sin x}{\cos x}\right)}\right) - \frac{{\varepsilon}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]
      unsub-neg [=>] 98.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(-\tan x\right) \cdot \left(\color{blue}{\left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right) - \frac{\varepsilon \cdot \sin x}{\cos x}\right)} - \frac{{\varepsilon}^{2} \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]

   if 2.0999999999999999e-11 < eps

   1. Initial program 53.0%

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

   2. Applied egg-rr 99.2%

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

      [Start] 53.0	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.2	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.1	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.28 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \tan x \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \frac{-0.3333333333333333}{\cos x} - {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) - \mathsf{fma}\left(\frac{\varepsilon}{\cos x}, \sin x, {\sin x}^{2} \cdot \frac{\varepsilon \cdot \varepsilon}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	45636

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	39433

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.25 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.1 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.75 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.1 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 4
Accuracy	77.2%
Cost	26116

\[\begin{array}{l} t_0 := \tan \varepsilon - \tan x\\ \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_0, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00135:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan x + t_0\\ \end{array} \]

Alternative 5
Accuracy	77.3%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00135:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan x + \left(\tan \varepsilon - \tan x\right)\\ \end{array} \]

Alternative 6
Accuracy	77.3%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00135:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7
Accuracy	58.1%
Cost	6464

\[\tan \varepsilon \]

Alternative 8
Accuracy	4.2%
Cost	64

\[0 \]

Alternative 9
Accuracy	30.9%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023136 
(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)))