Average Error: 37.0 → 0.3
Time: 11.0s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00021473042784006286:\\ \;\;\;\;\frac{t_0}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.0002116841733599596:\\ \;\;\;\;\begin{array}{l} t_1 := {\sin x}^{2}\\ t_2 := {\cos x}^{2}\\ t_3 := {\cos x}^{3}\\ t_4 := {\sin x}^{3}\\ \frac{{\varepsilon}^{2} \cdot t_4}{t_3} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{t_4 \cdot {\varepsilon}^{4}}{t_3} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_1}{t_2} + \left(\frac{\varepsilon \cdot t_1}{t_2} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \log \left(e^{\tan \varepsilon \cdot \frac{\sin x}{\cos x}}\right)} - \tan x\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := \tan x + \tan \varepsilon\\
\mathbf{if}\;\varepsilon \leq -0.00021473042784006286:\\
\;\;\;\;\frac{t_0}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 0.0002116841733599596:\\
\;\;\;\;\begin{array}{l}
t_1 := {\sin x}^{2}\\
t_2 := {\cos x}^{2}\\
t_3 := {\cos x}^{3}\\
t_4 := {\sin x}^{3}\\
\frac{{\varepsilon}^{2} \cdot t_4}{t_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{t_4 \cdot {\varepsilon}^{4}}{t_3} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_1}{t_2} + \left(\frac{\varepsilon \cdot t_1}{t_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)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0}{1 - \log \left(e^{\tan \varepsilon \cdot \frac{\sin x}{\cos x}}\right)} - \tan x\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -0.00021473042784006286)
     (- (/ t_0 (log (exp (- 1.0 (* (tan x) (tan eps)))))) (tan x))
     (if (<= eps 0.0002116841733599596)
       (let* ((t_1 (pow (sin x) 2.0))
              (t_2 (pow (cos x) 2.0))
              (t_3 (pow (cos x) 3.0))
              (t_4 (pow (sin x) 3.0)))
         (+
          (/ (* (pow eps 2.0) t_4) t_3)
          (+
           (/ (* (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_4 (pow eps 4.0)) t_3))
              (+
               (/ (* (pow eps 4.0) (pow (sin x) 5.0)) (pow (cos x) 5.0))
               (+
                (* 1.3333333333333333 (/ (* (pow eps 3.0) t_1) t_2))
                (+
                 (/ (* eps t_1) t_2)
                 (+
                  (* 0.6666666666666666 (/ (* (sin x) (pow eps 4.0)) (cos x)))
                  (* (pow eps 3.0) 0.3333333333333333)))))))))))
       (-
        (/ t_0 (- 1.0 (log (exp (* (tan eps) (/ (sin x) (cos x)))))))
        (tan x))))))
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 tmp;
	if (eps <= -0.00021473042784006286) {
		tmp = (t_0 / log(exp(1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else if (eps <= 0.0002116841733599596) {
		double t_1 = pow(sin(x), 2.0);
		double t_2 = pow(cos(x), 2.0);
		double t_3 = pow(cos(x), 3.0);
		double t_4 = pow(sin(x), 3.0);
		tmp = ((pow(eps, 2.0) * t_4) / t_3) + (((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_4 * pow(eps, 4.0)) / t_3)) + (((pow(eps, 4.0) * pow(sin(x), 5.0)) / pow(cos(x), 5.0)) + ((1.3333333333333333 * ((pow(eps, 3.0) * t_1) / t_2)) + (((eps * t_1) / t_2) + ((0.6666666666666666 * ((sin(x) * pow(eps, 4.0)) / cos(x))) + (pow(eps, 3.0) * 0.3333333333333333)))))))));
	} else {
		tmp = (t_0 / (1.0 - log(exp(tan(eps) * (sin(x) / cos(x)))))) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
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 < -2.1473042784006286e-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 add-log-exp_binary640.4

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

    4. Applied add-log-exp_binary640.4

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

    5. Applied diff-log_binary640.5

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

    6. Simplified0.4

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

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

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

    if -2.1473042784006286e-4 < eps < 2.1168417335995959e-4

    1. Initial program 44.6

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

    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 2.1168417335995959e-4 < eps

    1. Initial program 29.3

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

    2. Applied tan-sum_binary640.3

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

    3. Applied tan-quot_binary640.4

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

    4. Applied associate-*l/_binary640.4

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

    5. Applied add-log-exp_binary640.5

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

    6. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00021473042784006286:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.0002116841733599596:\\ \;\;\;\;\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan \varepsilon \cdot \frac{\sin x}{\cos x}}\right)} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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)))