Average Error: 37.1 → 1.0
Time: 11.5s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := \cos \varepsilon \cdot \cos x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7.034024772968383 \cdot 10^{-5}:\\ \;\;\;\;\begin{array}{l} t_2 := -\tan x\\ \mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\sin x \cdot \sin \varepsilon}{t_0}}, t_2\right) + \mathsf{fma}\left(t_2, 1, \tan x\right) \end{array}\\ \mathbf{elif}\;\varepsilon \leq 3.5100561387016338 \cdot 10^{-31}:\\ \;\;\;\;\begin{array}{l} t_3 := {\sin x}^{2}\\ t_4 := {\cos x}^{2}\\ \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\mathsf{fma}\left(\varepsilon, \frac{t_3}{t_4}, \varepsilon\right) + \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_3}{t_4}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_4}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - \sin \varepsilon \cdot \frac{\sin x}{t_0}} - \tan x\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := \cos \varepsilon \cdot \cos x\\
t_1 := \tan x + \tan \varepsilon\\
\mathbf{if}\;\varepsilon \leq -7.034024772968383 \cdot 10^{-5}:\\
\;\;\;\;\begin{array}{l}
t_2 := -\tan x\\
\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\sin x \cdot \sin \varepsilon}{t_0}}, t_2\right) + \mathsf{fma}\left(t_2, 1, \tan x\right)
\end{array}\\

\mathbf{elif}\;\varepsilon \leq 3.5100561387016338 \cdot 10^{-31}:\\
\;\;\;\;\begin{array}{l}
t_3 := {\sin x}^{2}\\
t_4 := {\cos x}^{2}\\
\left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\mathsf{fma}\left(\varepsilon, \frac{t_3}{t_4}, \varepsilon\right) + \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_3}{t_4}, {\varepsilon}^{3} \cdot 0.3333333333333333\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}{1 - \sin \varepsilon \cdot \frac{\sin x}{t_0}} - \tan x\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos eps) (cos x))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -7.034024772968383e-5)
     (let* ((t_2 (- (tan x))))
       (+
        (fma t_1 (/ 1.0 (- 1.0 (/ (* (sin x) (sin eps)) t_0))) t_2)
        (fma t_2 1.0 (tan x))))
     (if (<= eps 3.5100561387016338e-31)
       (let* ((t_3 (pow (sin x) 2.0)) (t_4 (pow (cos x) 2.0)))
         (+
          (+
           (/ (* (pow eps 3.0) (pow (sin x) 4.0)) (pow (cos x) 4.0))
           (+
            (fma eps (/ t_3 t_4) eps)
            (fma
             1.3333333333333333
             (/ (* (pow eps 3.0) t_3) t_4)
             (* (pow eps 3.0) 0.3333333333333333))))
          (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_4)))))
       (- (/ t_1 (- 1.0 (* (sin eps) (/ (sin x) 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 = cos(eps) * cos(x);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -7.034024772968383e-5) {
		double t_2_1 = -tan(x);
		tmp = fma(t_1, (1.0 / (1.0 - ((sin(x) * sin(eps)) / t_0))), t_2_1) + fma(t_2_1, 1.0, tan(x));
	} else if (eps <= 3.5100561387016338e-31) {
		double t_3 = pow(sin(x), 2.0);
		double t_4 = pow(cos(x), 2.0);
		tmp = (((pow(eps, 3.0) * pow(sin(x), 4.0)) / pow(cos(x), 4.0)) + (fma(eps, (t_3 / t_4), eps) + fma(1.3333333333333333, ((pow(eps, 3.0) * t_3) / t_4), (pow(eps, 3.0) * 0.3333333333333333)))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_4)));
	} else {
		tmp = (t_1 / (1.0 - (sin(eps) * (sin(x) / t_0)))) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.6
Herbie1.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -7.0340247729683834e-5

    1. Initial program 30.3

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

    2. Applied tan-sum_binary640.3

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

    3. Taylor expanded in x around inf 0.4

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

    4. Applied *-un-lft-identity_binary640.4

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

    5. Applied div-inv_binary640.4

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

    6. Applied prod-diff_binary640.4

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

    if -7.0340247729683834e-5 < eps < 3.5100561387016338e-31

    1. Initial program 45.2

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

    2. Applied tan-sum_binary6444.8

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

    3. Taylor expanded in eps around 0 0.2

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

    4. Simplified0.2

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

    if 3.5100561387016338e-31 < eps

    1. Initial program 30.1

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

    2. Applied tan-sum_binary642.8

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

    3. Taylor expanded in x around inf 2.8

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

    4. Applied associate-/l*_binary642.9

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

    5. Applied associate-/r/_binary642.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.034024772968383 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.5100561387016338 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right) + \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(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)))