Average Error: 37.0 → 14.9
Time: 32.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.2055148797080215 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.9017088514484585 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x \cdot \left(\tan x \cdot \tan x\right)}{\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} + \left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \tan x + \tan x \cdot \tan x\right)}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.2055148797080215 \cdot 10^{-18}:\\
\;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\

\mathbf{elif}\;\varepsilon \le 2.9017088514484585 \cdot 10^{-18}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\

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

\end{array}
double f(double x, double eps) {
        double r3662948 = x;
        double r3662949 = eps;
        double r3662950 = r3662948 + r3662949;
        double r3662951 = tan(r3662950);
        double r3662952 = tan(r3662948);
        double r3662953 = r3662951 - r3662952;
        return r3662953;
}
↓
double f(double x, double eps) {
        double r3662954 = eps;
        double r3662955 = -3.2055148797080215e-18;
        bool r3662956 = r3662954 <= r3662955;
        double r3662957 = x;
        double r3662958 = tan(r3662957);
        double r3662959 = r3662958 * r3662958;
        double r3662960 = tan(r3662954);
        double r3662961 = r3662960 * r3662960;
        double r3662962 = r3662959 - r3662961;
        double r3662963 = r3662958 - r3662960;
        double r3662964 = r3662962 / r3662963;
        double r3662965 = 1.0;
        double r3662966 = r3662960 * r3662958;
        double r3662967 = r3662965 - r3662966;
        double r3662968 = r3662964 / r3662967;
        double r3662969 = r3662968 - r3662958;
        double r3662970 = 2.9017088514484585e-18;
        bool r3662971 = r3662954 <= r3662970;
        double r3662972 = r3662957 * r3662954;
        double r3662973 = r3662957 + r3662954;
        double r3662974 = r3662972 * r3662973;
        double r3662975 = r3662974 + r3662954;
        double r3662976 = r3662960 + r3662958;
        double r3662977 = sin(r3662957);
        double r3662978 = r3662977 * r3662960;
        double r3662979 = cos(r3662957);
        double r3662980 = r3662978 / r3662979;
        double r3662981 = r3662965 - r3662980;
        double r3662982 = r3662976 / r3662981;
        double r3662983 = r3662982 * r3662982;
        double r3662984 = r3662983 * r3662982;
        double r3662985 = r3662958 * r3662959;
        double r3662986 = r3662984 - r3662985;
        double r3662987 = r3662982 * r3662958;
        double r3662988 = r3662987 + r3662959;
        double r3662989 = r3662983 + r3662988;
        double r3662990 = r3662986 / r3662989;
        double r3662991 = r3662971 ? r3662975 : r3662990;
        double r3662992 = r3662956 ? r3662969 : r3662991;
        return r3662992;
}

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
Target15.4
Herbie14.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -3.2055148797080215e-18

    1. Initial program 29.3

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

    3. Applied tan-sum1.2

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

    5. Applied flip-+1.2

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

    if -3.2055148797080215e-18 < eps < 2.9017088514484585e-18

    1. Initial program 45.5

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

    3. Applied tan-sum45.5

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

    5. Applied tan-quot45.5

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

    6. Applied associate-*l/45.5

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

    7. Taylor expanded around 0 30.8

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

    8. Simplified30.8

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

    if 2.9017088514484585e-18 < eps

    1. Initial program 30.1

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

    3. Applied tan-sum1.0

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

    5. Applied tan-quot1.0

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

    6. Applied associate-*l/1.0

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

    8. Applied flip3--1.2

      \[\leadsto \color{blue}{\frac{{\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right)}^{3} - {\left(\tan x\right)}^{3}}{\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} + \left(\tan x \cdot \tan x + \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \tan x\right)}}\]

    9. Simplified1.1

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

  4. Final simplification14.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.2055148797080215 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.9017088514484585 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x \cdot \left(\tan x \cdot \tan x\right)}{\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} + \left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \tan x + \tan x \cdot \tan x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))