Average Error: 37.6 → 15.7
Time: 11.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.6784667043225302 \cdot 10^{-32}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.04782928541883701 \cdot 10^{-23}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.6784667043225302 \cdot 10^{-32}:\\
\;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.04782928541883701 \cdot 10^{-23}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\

\end{array}
double f(double x, double eps) {
        double r103812 = x;
        double r103813 = eps;
        double r103814 = r103812 + r103813;
        double r103815 = tan(r103814);
        double r103816 = tan(r103812);
        double r103817 = r103815 - r103816;
        return r103817;
}
↓
double f(double x, double eps) {
        double r103818 = eps;
        double r103819 = -4.67846670432253e-32;
        bool r103820 = r103818 <= r103819;
        double r103821 = x;
        double r103822 = tan(r103821);
        double r103823 = tan(r103818);
        double r103824 = r103822 + r103823;
        double r103825 = 1.0;
        double r103826 = r103822 * r103823;
        double r103827 = r103825 + r103826;
        double r103828 = r103824 * r103827;
        double r103829 = cos(r103821);
        double r103830 = r103828 * r103829;
        double r103831 = r103826 * r103826;
        double r103832 = r103825 - r103831;
        double r103833 = sin(r103821);
        double r103834 = r103832 * r103833;
        double r103835 = r103830 - r103834;
        double r103836 = r103832 * r103829;
        double r103837 = r103835 / r103836;
        double r103838 = 1.047829285418837e-23;
        bool r103839 = r103818 <= r103838;
        double r103840 = r103821 * r103818;
        double r103841 = r103818 + r103821;
        double r103842 = r103840 * r103841;
        double r103843 = r103842 + r103818;
        double r103844 = 2.0;
        double r103845 = pow(r103833, r103844);
        double r103846 = sin(r103818);
        double r103847 = pow(r103846, r103844);
        double r103848 = r103845 * r103847;
        double r103849 = cos(r103818);
        double r103850 = pow(r103849, r103844);
        double r103851 = pow(r103829, r103844);
        double r103852 = r103850 * r103851;
        double r103853 = r103848 / r103852;
        double r103854 = r103825 - r103853;
        double r103855 = r103824 / r103854;
        double r103856 = r103855 * r103827;
        double r103857 = r103856 - r103822;
        double r103858 = r103839 ? r103843 : r103857;
        double r103859 = r103820 ? r103837 : r103858;
        return r103859;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie15.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.67846670432253e-32

    1. Initial program 31.0

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    2. Using strategy rm

    3. Applied tan-sum2.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied flip--2.8

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

    6. Applied associate-/r/2.8

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

    7. Simplified2.8

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

    9. Applied tan-quot2.9

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

    10. Applied associate-*l/2.9

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

    11. Applied frac-sub2.9

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

    if -4.67846670432253e-32 < eps < 1.047829285418837e-23

    1. Initial program 46.1

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    2. Using strategy rm

    3. Applied tan-sum46.1

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

    4. Taylor expanded around 0 32.3

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

    5. Simplified32.0

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

    if 1.047829285418837e-23 < eps

    1. Initial program 30.1

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    2. Using strategy rm

    3. Applied tan-sum1.6

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied flip--1.6

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

    6. Applied associate-/r/1.6

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

    7. Simplified1.6

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

    8. Taylor expanded around inf 1.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.6784667043225302 \cdot 10^{-32}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.04782928541883701 \cdot 10^{-23}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot {\left(\cos x\right)}^{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))