2tan (problem 3.3.2)

Average Accuracy: 42.1% → 98.9%

Time: 26.7s

Precision: binary64

Cost: 72392

Math

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

↓

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

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))))
   (if (<= eps -1.28e+16)
     (-
      (/
       (* t_1 (fma (tan x) (tan eps) 1.0))
       (- 1.0 (* (tan eps) (* (tan eps) (pow (tan x) 2.0)))))
      (tan x))
     (if (<= eps 2.1e-11)
       (+
        (/
         (* (tan x) (* (sin eps) (/ (tan x) (cos eps))))
         (- 1.0 (* t_0 (tan x))))
        (/ t_0 (- 1.0 (/ (* eps (sin x)) (cos x)))))
       (fma (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_1 (- (tan x)))))))

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 tmp;
	if (eps <= -1.28e+16) {
		tmp = ((t_1 * fma(tan(x), tan(eps), 1.0)) / (1.0 - (tan(eps) * (tan(eps) * pow(tan(x), 2.0))))) - tan(x);
	} else if (eps <= 2.1e-11) {
		tmp = ((tan(x) * (sin(eps) * (tan(x) / cos(eps)))) / (1.0 - (t_0 * tan(x)))) + (t_0 / (1.0 - ((eps * sin(x)) / cos(x))));
	} else {
		tmp = fma((1.0 / (1.0 - (tan(x) * tan(eps)))), t_1, -tan(x));
	}
	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))
	tmp = 0.0
	if (eps <= -1.28e+16)
		tmp = Float64(Float64(Float64(t_1 * fma(tan(x), tan(eps), 1.0)) / Float64(1.0 - Float64(tan(eps) * Float64(tan(eps) * (tan(x) ^ 2.0))))) - tan(x));
	elseif (eps <= 2.1e-11)
		tmp = Float64(Float64(Float64(tan(x) * Float64(sin(eps) * Float64(tan(x) / cos(eps)))) / Float64(1.0 - Float64(t_0 * tan(x)))) + Float64(t_0 / Float64(1.0 - Float64(Float64(eps * sin(x)) / cos(x)))));
	else
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_1, Float64(-tan(x)));
	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]}, If[LessEqual[eps, -1.28e+16], N[(N[(N[(t$95$1 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.1e-11], N[(N[(N[(N[Tan[x], $MachinePrecision] * N[(N[Sin[eps], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$0 * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(1.0 - N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + (-N[Tan[x], $MachinePrecision])), $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 99.4%

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

      [Start] 54.3	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      flip-- [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x \]
      associate-/r/ [=>] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      metadata-eval [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{1} - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   3. Simplified 99.4%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      associate-*l/ [=>] 99.4	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}} - \tan x \]
      +-commutative [=>] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \color{blue}{\left(\tan x \cdot \tan \varepsilon + 1\right)}}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x \]
      fma-def [=>] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x \]
      associate-*r* [=>] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \color{blue}{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \tan x\right) \cdot \tan \varepsilon}} - \tan x \]
      *-commutative [=>] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \color{blue}{\tan \varepsilon \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \tan x\right)}} - \tan x \]
      *-commutative [=>] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \tan \varepsilon \cdot \left(\color{blue}{\left(\tan \varepsilon \cdot \tan x\right)} \cdot \tan x\right)} - \tan x \]
      associate-*l* [=>] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \tan \varepsilon \cdot \color{blue}{\left(\tan \varepsilon \cdot \left(\tan x \cdot \tan x\right)\right)}} - \tan x \]
      unpow2 [<=] 99.4	\[ \frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \tan \varepsilon \cdot \left(\tan \varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - \tan x \]

   if -1.28e16 < eps < 2.0999999999999999e-11

   1. Initial program 30.6%

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

   2. Applied egg-rr 32.1%

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

      [Start] 30.6	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 32.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      add-sqr-sqrt [=>] 32.1	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\sqrt{1 - \tan x \cdot \tan \varepsilon} \cdot \sqrt{1 - \tan x \cdot \tan \varepsilon}}} - \tan x \]
      associate-/r* [=>] 32.1	\[ \color{blue}{\frac{\frac{\tan x + \tan \varepsilon}{\sqrt{1 - \tan x \cdot \tan \varepsilon}}}{\sqrt{1 - \tan x \cdot \tan \varepsilon}}} - \tan x \]

   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%

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

   5. Applied egg-rr 89.5%

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

      [Start] 61.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} - \frac{\sin x}{\cos x}\right) \]
      clear-num [=>] 55.2	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} - \color{blue}{\frac{1}{\frac{\cos x}{\sin x}}}\right) \]
      frac-sub [=>] 55.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \color{blue}{\frac{\frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} - \left(1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}\right) \cdot 1}{\left(1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}\right) \cdot \frac{\cos x}{\sin x}}} \]

   6. Simplified 99.7%

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

      [Start] 89.5	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \frac{\left(\tan x \cdot \frac{1}{\tan x} - 1\right) + \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x}{\frac{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}}{\tan x}} \]
      associate-/r/ [=>] 89.6	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \color{blue}{\frac{\left(\tan x \cdot \frac{1}{\tan x} - 1\right) + \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x}{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}} \cdot \tan x} \]
      +-commutative [=>] 89.6	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \frac{\color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x + \left(\tan x \cdot \frac{1}{\tan x} - 1\right)}}{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}} \cdot \tan x \]
      rgt-mult-inverse [=>] 99.7	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \frac{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x + \left(\color{blue}{1} - 1\right)}{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}} \cdot \tan x \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \frac{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x + \color{blue}{0}}{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}} \cdot \tan x \]
      +-rgt-identity [=>] 99.7	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \frac{\color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x}}{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}} \cdot \tan x \]
      associate-*l/ [=>] 99.7	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon}{\frac{\sin \varepsilon}{\cos x}}}} + \color{blue}{\frac{\left(\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \tan x\right) \cdot \tan x}{1 + \frac{-1}{\frac{\frac{\cos \varepsilon}{\sin \varepsilon}}{\tan x}}}} \]

   7. Taylor expanded in eps around 0 98.6%

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

   if 2.0999999999999999e-11 < eps

   1. Initial program 53.0%

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

   2. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      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	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-commutative [=>] 99.1	\[ \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]

   3. Applied egg-rr 99.2%

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

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

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	78528

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

Alternative 2
Accuracy	99.4%
Cost	65284

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \tan \varepsilon \cdot \left(\tan \varepsilon \cdot {\tan x}^{2}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_1}, t_0, -\tan x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	46088

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;t_0 - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_2}, t_1, t_0\right)\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	39433

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.1 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	39172

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -8.4 \cdot 10^{-10}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.1 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7
Accuracy	77.3%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00135\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 8
Accuracy	77.3%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00135\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 9
Accuracy	58.0%
Cost	12992

\[\frac{\sin \varepsilon}{\cos \varepsilon} \]

Alternative 10
Accuracy	39.8%
Cost	6720

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

Alternative 11
Accuracy	3.6%
Cost	128

\[-x \]

Alternative 12
Accuracy	4.2%
Cost	64

\[0 \]

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)))