Average Error: 36.7 → 0.3
Time: 12.4s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := {\sin x}^{3}\\ t_1 := {\cos x}^{2}\\ t_2 := {\cos x}^{3}\\ t_3 := \tan x \cdot \tan \varepsilon\\ t_4 := \tan x + \tan \varepsilon\\ t_5 := {\sin x}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.00020625213391552245:\\ \;\;\;\;\frac{t_4}{1 - t_3} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.687244365865793 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\varepsilon}^{2} \cdot t_0}{t_2} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{t_0 \cdot {\varepsilon}^{4}}{t_2} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_5}{t_1} + \left(\frac{\varepsilon \cdot t_5}{t_1} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_4, \frac{1}{1 - \sqrt[3]{{t_3}^{3}}}, -\tan x\right)\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := {\sin x}^{3}\\
t_1 := {\cos x}^{2}\\
t_2 := {\cos x}^{3}\\
t_3 := \tan x \cdot \tan \varepsilon\\
t_4 := \tan x + \tan \varepsilon\\
t_5 := {\sin x}^{2}\\
\mathbf{if}\;\varepsilon \leq -0.00020625213391552245:\\
\;\;\;\;\frac{t_4}{1 - t_3} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 6.687244365865793 \cdot 10^{-5}:\\
\;\;\;\;\frac{{\varepsilon}^{2} \cdot t_0}{t_2} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{t_0 \cdot {\varepsilon}^{4}}{t_2} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_5}{t_1} + \left(\frac{\varepsilon \cdot t_5}{t_1} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t_4, \frac{1}{1 - \sqrt[3]{{t_3}^{3}}}, -\tan x\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 3.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (pow (cos x) 3.0))
        (t_3 (* (tan x) (tan eps)))
        (t_4 (+ (tan x) (tan eps)))
        (t_5 (pow (sin x) 2.0)))
   (if (<= eps -0.00020625213391552245)
     (- (/ t_4 (- 1.0 t_3)) (tan x))
     (if (<= eps 6.687244365865793e-5)
       (+
        (/ (* (pow eps 2.0) t_0) t_2)
        (+
         (/ (* (pow eps 2.0) (sin x)) (cos x))
         (+
          eps
          (+
           (/ (* (pow eps 3.0) (pow (sin x) 4.0)) (pow (cos x) 4.0))
           (+
            (* 1.6666666666666667 (/ (* t_0 (pow eps 4.0)) t_2))
            (+
             (/ (* (pow eps 4.0) (pow (sin x) 5.0)) (pow (cos x) 5.0))
             (+
              (* 1.3333333333333333 (/ (* (pow eps 3.0) t_5) t_1))
              (+
               (/ (* eps t_5) t_1)
               (+
                (* 0.6666666666666666 (/ (* (sin x) (pow eps 4.0)) (cos x)))
                (* (pow eps 3.0) 0.3333333333333333))))))))))
       (fma t_4 (/ 1.0 (- 1.0 (cbrt (pow t_3 3.0)))) (- (tan x)))))))
double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}
↓
double code(double x, double eps) {
	double t_0 = pow(sin(x), 3.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = pow(cos(x), 3.0);
	double t_3 = tan(x) * tan(eps);
	double t_4 = tan(x) + tan(eps);
	double t_5 = pow(sin(x), 2.0);
	double tmp;
	if (eps <= -0.00020625213391552245) {
		tmp = (t_4 / (1.0 - t_3)) - tan(x);
	} else if (eps <= 6.687244365865793e-5) {
		tmp = ((pow(eps, 2.0) * t_0) / t_2) + (((pow(eps, 2.0) * sin(x)) / cos(x)) + (eps + (((pow(eps, 3.0) * pow(sin(x), 4.0)) / pow(cos(x), 4.0)) + ((1.6666666666666667 * ((t_0 * pow(eps, 4.0)) / t_2)) + (((pow(eps, 4.0) * pow(sin(x), 5.0)) / pow(cos(x), 5.0)) + ((1.3333333333333333 * ((pow(eps, 3.0) * t_5) / t_1)) + (((eps * t_5) / t_1) + ((0.6666666666666666 * ((sin(x) * pow(eps, 4.0)) / cos(x))) + (pow(eps, 3.0) * 0.3333333333333333)))))))));
	} else {
		tmp = fma(t_4, (1.0 / (1.0 - cbrt(pow(t_3, 3.0)))), -tan(x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.0
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.06252133915522446e-4

    1. Initial program 29.1

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

    2. Applied egg-rr0.4

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

    if -2.06252133915522446e-4 < eps < 6.687244365865793e-5

    1. Initial program 44.2

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

    2. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{{\varepsilon}^{4} \cdot \sin x}{\cos x} + 0.3333333333333333 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\right)\right)\right)\right)} \]

    if 6.687244365865793e-5 < eps

    1. Initial program 29.6

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

    2. Applied egg-rr0.4

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

    3. Applied egg-rr0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00020625213391552245:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.687244365865793 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}, -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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)))