Average Error: 37.1 → 0.3
Time: 20.8s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00020782601609953486:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - t_1 \cdot t_1} \cdot \left(1 + t_1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00044940075034421095:\\ \;\;\;\;\begin{array}{l} t_2 := {\sin x}^{3}\\ t_3 := {\sin x}^{2}\\ t_4 := {\cos x}^{2}\\ t_5 := {\cos x}^{3}\\ \frac{{\varepsilon}^{2} \cdot t_2}{t_5} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{t_2 \cdot {\varepsilon}^{4}}{t_5} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_3}{t_4} + \left(\frac{\varepsilon \cdot t_3}{t_4} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, t_1 + -1, t_0 \cdot \cos x\right)}{\cos x \cdot \left(1 - t_1\right)}\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := \tan x + \tan \varepsilon\\
t_1 := \tan x \cdot \tan \varepsilon\\
\mathbf{if}\;\varepsilon \leq -0.00020782601609953486:\\
\;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - t_1 \cdot t_1} \cdot \left(1 + t_1\right), -\tan x\right)\\

\mathbf{elif}\;\varepsilon \leq 0.00044940075034421095:\\
\;\;\;\;\begin{array}{l}
t_2 := {\sin x}^{3}\\
t_3 := {\sin x}^{2}\\
t_4 := {\cos x}^{2}\\
t_5 := {\cos x}^{3}\\
\frac{{\varepsilon}^{2} \cdot t_2}{t_5} + \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_2 \cdot {\varepsilon}^{4}}{t_5} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_3}{t_4} + \left(\frac{\varepsilon \cdot t_3}{t_4} + \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)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sin x, t_1 + -1, t_0 \cdot \cos x\right)}{\cos x \cdot \left(1 - t_1\right)}\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (* (tan x) (tan eps))))
   (if (<= eps -0.00020782601609953486)
     (fma t_0 (* (/ 1.0 (- 1.0 (* t_1 t_1))) (+ 1.0 t_1)) (- (tan x)))
     (if (<= eps 0.00044940075034421095)
       (let* ((t_2 (pow (sin x) 3.0))
              (t_3 (pow (sin x) 2.0))
              (t_4 (pow (cos x) 2.0))
              (t_5 (pow (cos x) 3.0)))
         (+
          (/ (* (pow eps 2.0) t_2) t_5)
          (+
           (/ (* (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_2 (pow eps 4.0)) t_5))
              (+
               (/ (* (pow eps 4.0) (pow (sin x) 5.0)) (pow (cos x) 5.0))
               (+
                (* 1.3333333333333333 (/ (* (pow eps 3.0) t_3) t_4))
                (+
                 (/ (* eps t_3) t_4)
                 (+
                  (* 0.6666666666666666 (/ (* (sin x) (pow eps 4.0)) (cos x)))
                  (* (pow eps 3.0) 0.3333333333333333)))))))))))
       (/
        (fma (sin x) (+ t_1 -1.0) (* t_0 (cos x)))
        (* (cos x) (- 1.0 t_1)))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = tan(x) * tan(eps);
	double tmp;
	if (eps <= -0.00020782601609953486) {
		tmp = fma(t_0, ((1.0 / (1.0 - (t_1 * t_1))) * (1.0 + t_1)), -tan(x));
	} else if (eps <= 0.00044940075034421095) {
		double t_2 = pow(sin(x), 3.0);
		double t_3 = pow(sin(x), 2.0);
		double t_4 = pow(cos(x), 2.0);
		double t_5 = pow(cos(x), 3.0);
		tmp = ((pow(eps, 2.0) * t_2) / t_5) + (((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_2 * pow(eps, 4.0)) / t_5)) + (((pow(eps, 4.0) * pow(sin(x), 5.0)) / pow(cos(x), 5.0)) + ((1.3333333333333333 * ((pow(eps, 3.0) * t_3) / t_4)) + (((eps * t_3) / t_4) + ((0.6666666666666666 * ((sin(x) * pow(eps, 4.0)) / cos(x))) + (pow(eps, 3.0) * 0.3333333333333333)))))))));
	} else {
		tmp = fma(sin(x), (t_1 + -1.0), (t_0 * cos(x))) / (cos(x) * (1.0 - t_1));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.1
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.07826016099534858e-4

    1. Initial program 29.9

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

    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_binary640.3

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

    4. Applied fma-neg_binary640.3

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

    5. Applied flip--_binary640.4

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

    6. Applied associate-/r/_binary640.4

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

    7. Simplified0.4

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

    if -2.07826016099534858e-4 < eps < 4.49400750344210946e-4

    1. Initial program 44.7

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

    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 4.49400750344210946e-4 < eps

    1. Initial program 29.5

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

    2. Applied tan-quot_binary6429.4

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

    3. Applied tan-sum_binary640.4

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

    4. Applied frac-sub_binary640.4

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

    5. Simplified0.4

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

    6. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00020782601609953486:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00044940075034421095:\\ \;\;\;\;\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \tan x \cdot \tan \varepsilon + -1, \left(\tan x + \tan \varepsilon\right) \cdot \cos x\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(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)))