2tan (problem 3.3.2)

Average Accuracy: 42.5% → 99.4%

Time: 21.9s

Precision: binary64

Cost: 59144

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(t_2, \frac{\sin \varepsilon}{\cos \varepsilon}, -1\right)}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(t_2 + {t_2}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (/ (sin x) (cos x))))
   (if (<= eps -1.3e-7)
     (fma t_1 (/ -1.0 (fma t_2 (/ (sin eps) (cos eps)) -1.0)) t_0)
     (if (<= eps 1.85e-7)
       (*
        eps
        (+
         (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
         (* eps (+ t_2 (pow t_2 3.0)))))
       (fma t_1 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_0)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = tan(x) + tan(eps);
	double t_2 = sin(x) / cos(x);
	double tmp;
	if (eps <= -1.3e-7) {
		tmp = fma(t_1, (-1.0 / fma(t_2, (sin(eps) / cos(eps)), -1.0)), t_0);
	} else if (eps <= 1.85e-7) {
		tmp = eps * ((1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))) + (eps * (t_2 + pow(t_2, 3.0))));
	} else {
		tmp = fma(t_1, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_0);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = Float64(tan(x) + tan(eps))
	t_2 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -1.3e-7)
		tmp = fma(t_1, Float64(-1.0 / fma(t_2, Float64(sin(eps) / cos(eps)), -1.0)), t_0);
	elseif (eps <= 1.85e-7)
		tmp = Float64(eps * Float64(Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + Float64(eps * Float64(t_2 + (t_2 ^ 3.0)))));
	else
		tmp = fma(t_1, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_0);
	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[Tan[x], $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]}, If[LessEqual[eps, -1.3e-7], N[(t$95$1 * N[(-1.0 / N[(t$95$2 * N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[eps, 1.85e-7], N[(eps * N[(N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(t$95$2 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Target

Original	42.5%
Target	75.4%
Herbie	99.4%

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.29999999999999999e-7

   1. Initial program 53.2%

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

   2. Applied egg-rr 99.4%

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

      [Start] 53.2	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      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 [=>] 99.4	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 99.3%

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

      [Start] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      tan-quot [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}, -\tan x\right) \]
      associate-*r/ [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}}, -\tan x\right) \]

   4. Simplified 99.4%

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

      [Start] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right) \]
      *-commutative [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\color{blue}{\sin \varepsilon \cdot \tan x}}{\cos \varepsilon}}, -\tan x\right) \]
      associate-/l* [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}}}, -\tan x\right) \]

   5. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}}, -\tan x\right) \]
      frac-2neg [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{-1}{-\left(1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)}}, -\tan x\right) \]
      metadata-eval [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{-1}}{-\left(1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)}, -\tan x\right) \]
      div-inv [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{-1 \cdot \frac{1}{-\left(1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)}}, -\tan x\right) \]
      sub-neg [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-\color{blue}{\left(1 + \left(-\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)\right)}}, -\tan x\right) \]
      distribute-neg-in [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{\color{blue}{\left(-1\right) + \left(-\left(-\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)\right)}}, -\tan x\right) \]
      metadata-eval [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{\color{blue}{-1} + \left(-\left(-\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)\right)}, -\tan x\right) \]
      associate-/r/ [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}\right)\right)}, -\tan x\right) \]
      tan-quot [<=] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\left(-\color{blue}{\tan \varepsilon} \cdot \tan x\right)\right)}, -\tan x\right) \]
      distribute-rgt-neg-in [=>] 99.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\color{blue}{\tan \varepsilon \cdot \left(-\tan x\right)}\right)}, -\tan x\right) \]
      add-sqr-sqrt [=>] 49.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\left(\sqrt{-\tan x} \cdot \sqrt{-\tan x}\right)}\right)}, -\tan x\right) \]
      sqrt-unprod [=>] 77.6	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\sqrt{\left(-\tan x\right) \cdot \left(-\tan x\right)}}\right)}, -\tan x\right) \]
      sqr-neg [=>] 77.6	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \sqrt{\color{blue}{\tan x \cdot \tan x}}\right)}, -\tan x\right) \]
      sqrt-unprod [<=] 28.1	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\left(\sqrt{\tan x} \cdot \sqrt{\tan x}\right)}\right)}, -\tan x\right) \]
      add-sqr-sqrt [<=] 56.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\tan x}\right)}, -\tan x\right) \]
      tan-quot [=>] 56.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x\right)}, -\tan x\right) \]

   6. Simplified 99.4%

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

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

   7. Taylor expanded in x around inf 99.3%

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

   8. Simplified 99.3%

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

      [Start] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1}, -\tan x\right) \]
      *-commutative [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\frac{\sin x \cdot \sin \varepsilon}{\color{blue}{\cos x \cdot \cos \varepsilon}} - 1}, -\tan x\right) \]
      times-frac [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\color{blue}{\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - 1}, -\tan x\right) \]
      fma-neg [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, -1\right)}}, -\tan x\right) \]
      metadata-eval [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, \color{blue}{-1}\right)}, -\tan x\right) \]

   if -1.29999999999999999e-7 < eps < 1.85000000000000002e-7

   1. Initial program 31.0%

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

   2. Applied egg-rr 31.6%

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

      [Start] 31.0	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 31.6	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 31.6	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 31.6	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Taylor expanded in eps around 0 99.6%

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

   4. Simplified 99.6%

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

      [Start] 99.6	\[ \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      +-commutative [=>] 99.6	\[ \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2}} \]
      *-commutative [=>] 99.6	\[ \color{blue}{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2} \]
      unpow2 [=>] 99.6	\[ \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]
      associate-*r* [=>] 99.6	\[ \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon + \color{blue}{\left(\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot \varepsilon\right) \cdot \varepsilon} \]
      distribute-rgt-out [=>] 99.6	\[ \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot \varepsilon\right)} \]

   if 1.85000000000000002e-7 < eps

   1. Initial program 52.8%

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

   2. Applied egg-rr 99.3%

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

      [Start] 52.8	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.3	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.3	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.3	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 99.3%

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

      [Start] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      tan-quot [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}, -\tan x\right) \]
      associate-*r/ [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}}, -\tan x\right) \]

   4. Simplified 99.3%

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

      [Start] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right) \]
      *-commutative [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\color{blue}{\sin \varepsilon \cdot \tan x}}{\cos \varepsilon}}, -\tan x\right) \]
      associate-/l* [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}}}, -\tan x\right) \]

   5. Applied egg-rr 99.3%

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

      [Start] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}}, -\tan x\right) \]
      frac-2neg [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{-1}{-\left(1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)}}, -\tan x\right) \]
      metadata-eval [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{-1}}{-\left(1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)}, -\tan x\right) \]
      div-inv [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{-1 \cdot \frac{1}{-\left(1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)}}, -\tan x\right) \]
      sub-neg [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-\color{blue}{\left(1 + \left(-\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)\right)}}, -\tan x\right) \]
      distribute-neg-in [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{\color{blue}{\left(-1\right) + \left(-\left(-\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)\right)}}, -\tan x\right) \]
      metadata-eval [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{\color{blue}{-1} + \left(-\left(-\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}\right)\right)}, -\tan x\right) \]
      associate-/r/ [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}\right)\right)}, -\tan x\right) \]
      tan-quot [<=] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\left(-\color{blue}{\tan \varepsilon} \cdot \tan x\right)\right)}, -\tan x\right) \]
      distribute-rgt-neg-in [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\color{blue}{\tan \varepsilon \cdot \left(-\tan x\right)}\right)}, -\tan x\right) \]
      add-sqr-sqrt [=>] 48.1	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\left(\sqrt{-\tan x} \cdot \sqrt{-\tan x}\right)}\right)}, -\tan x\right) \]
      sqrt-unprod [=>] 76.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\sqrt{\left(-\tan x\right) \cdot \left(-\tan x\right)}}\right)}, -\tan x\right) \]
      sqr-neg [=>] 76.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \sqrt{\color{blue}{\tan x \cdot \tan x}}\right)}, -\tan x\right) \]
      sqrt-unprod [<=] 28.2	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\left(\sqrt{\tan x} \cdot \sqrt{\tan x}\right)}\right)}, -\tan x\right) \]
      add-sqr-sqrt [<=] 55.7	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\tan \varepsilon \cdot \color{blue}{\tan x}\right)}, -\tan x\right) \]
      tan-quot [=>] 55.7	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x\right)}, -\tan x\right) \]

   6. Simplified 99.3%

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

      [Start] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, -1 \cdot \frac{1}{-1 + \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      associate-*r/ [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\frac{-1 \cdot 1}{-1 + \tan x \cdot \tan \varepsilon}}, -\tan x\right) \]
      metadata-eval [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{\color{blue}{-1}}{-1 + \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      +-commutative [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}}, -\tan x\right) \]
      fma-def [=>] 99.3	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}}, -\tan x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, -1\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	58628

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, -1\right)}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	45705

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	39433

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.6 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5
Accuracy	76.5%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.36 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 6
Accuracy	76.6%
Cost	26116

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00068:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \tan x, \tan x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 7
Accuracy	76.5%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \tan x, \tan x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 8
Accuracy	76.5%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 9
Accuracy	76.5%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 10
Accuracy	57.5%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 8.2 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \]

Alternative 11
Accuracy	57.8%
Cost	12992

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

Alternative 12
Accuracy	30.6%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023152 
(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)))