Average Error: 36.6 → 0.3
Time: 14.4s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00021560437678393656:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1 - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00031230127715924045:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{\cos x}{\sin x}\\ t_3 := {\sin x}^{2}\\ t_4 := {\cos x}^{2}\\ \left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{t_2}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{t_2}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_3}{t_4}, \mathsf{fma}\left(\varepsilon, \frac{t_3}{t_4}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_4}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} - \tan x\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := 1 - \tan x \cdot \tan \varepsilon\\
t_1 := \tan x + \tan \varepsilon\\
\mathbf{if}\;\varepsilon \leq -0.00021560437678393656:\\
\;\;\;\;\frac{1}{t_0} \cdot t_1 - \tan x\\

\mathbf{elif}\;\varepsilon \leq 0.00031230127715924045:\\
\;\;\;\;\begin{array}{l}
t_2 := \frac{\cos x}{\sin x}\\
t_3 := {\sin x}^{2}\\
t_4 := {\cos x}^{2}\\
\left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{t_2}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{t_2}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_3}{t_4}, \mathsf{fma}\left(\varepsilon, \frac{t_3}{t_4}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_4}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_1}{t_0} - \tan x\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps)))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -0.00021560437678393656)
     (- (* (/ 1.0 t_0) t_1) (tan x))
     (if (<= eps 0.00031230127715924045)
       (let* ((t_2 (/ (cos x) (sin x)))
              (t_3 (pow (sin x) 2.0))
              (t_4 (pow (cos x) 2.0)))
         (+
          (+
           eps
           (+
            (/ (pow eps 3.0) (pow t_2 4.0))
            (fma
             1.6666666666666667
             (/ (pow eps 4.0) (pow t_2 3.0))
             (fma
              (/ (pow eps 4.0) (pow (cos x) 5.0))
              (pow (sin x) 5.0)
              (fma
               1.3333333333333333
               (/ (* (pow eps 3.0) t_3) t_4)
               (fma
                eps
                (/ t_3 t_4)
                (fma
                 0.3333333333333333
                 (pow eps 3.0)
                 (*
                  0.6666666666666666
                  (* (pow eps 4.0) (/ (sin x) (cos x)))))))))))
          (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_4)))))
       (- (/ t_1 t_0) (tan x))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -0.00021560437678393656) {
		tmp = ((1.0 / t_0) * t_1) - tan(x);
	} else if (eps <= 0.00031230127715924045) {
		double t_2 = cos(x) / sin(x);
		double t_3 = pow(sin(x), 2.0);
		double t_4 = pow(cos(x), 2.0);
		tmp = (eps + ((pow(eps, 3.0) / pow(t_2, 4.0)) + fma(1.6666666666666667, (pow(eps, 4.0) / pow(t_2, 3.0)), fma((pow(eps, 4.0) / pow(cos(x), 5.0)), pow(sin(x), 5.0), fma(1.3333333333333333, ((pow(eps, 3.0) * t_3) / t_4), fma(eps, (t_3 / t_4), fma(0.3333333333333333, pow(eps, 3.0), (0.6666666666666666 * (pow(eps, 4.0) * (sin(x) / cos(x))))))))))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_4)));
	} else {
		tmp = (t_1 / t_0) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.6
Target15.2
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.1560437678393656e-4

    1. Initial program 29.5

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

    2. Applied tan-sum_binary640.3

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

    3. Applied div-inv_binary640.4

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

    4. Applied *-commutative_binary640.4

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

    if -2.1560437678393656e-4 < eps < 3.12301277159240454e-4

    1. Initial program 44.2

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

    2. Taylor expanded in eps around 0 0.2

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

    3. Simplified0.2

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

    if 3.12301277159240454e-4 < eps

    1. Initial program 29.5

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

    2. Applied tan-sum_binary640.3

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

    3. Applied +-commutative_binary640.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00021560437678393656:\\ \;\;\;\;\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00031230127715924045:\\ \;\;\;\;\left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2021357 
(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)))