Average Error: 37.0 → 15.0
Time: 30.2s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.110688788355602 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.0915005808183093 \cdot 10^{-33}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.110688788355602 \cdot 10^{-13}:\\
\;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\

\mathbf{elif}\;\varepsilon \le 1.0915005808183093 \cdot 10^{-33}:\\
\;\;\;\;\varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right) + \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\

\end{array}
double f(double x, double eps) {
        double r3345940 = x;
        double r3345941 = eps;
        double r3345942 = r3345940 + r3345941;
        double r3345943 = tan(r3345942);
        double r3345944 = tan(r3345940);
        double r3345945 = r3345943 - r3345944;
        return r3345945;
}
↓
double f(double x, double eps) {
        double r3345946 = eps;
        double r3345947 = -8.110688788355602e-13;
        bool r3345948 = r3345946 <= r3345947;
        double r3345949 = 1.0;
        double r3345950 = x;
        double r3345951 = tan(r3345950);
        double r3345952 = tan(r3345946);
        double r3345953 = r3345951 * r3345952;
        double r3345954 = r3345949 - r3345953;
        double r3345955 = r3345951 + r3345952;
        double r3345956 = r3345954 / r3345955;
        double r3345957 = r3345949 / r3345956;
        double r3345958 = r3345957 - r3345951;
        double r3345959 = 1.0915005808183093e-33;
        bool r3345960 = r3345946 <= r3345959;
        double r3345961 = r3345946 + r3345950;
        double r3345962 = r3345950 * r3345961;
        double r3345963 = r3345946 * r3345962;
        double r3345964 = r3345963 + r3345946;
        double r3345965 = r3345960 ? r3345964 : r3345958;
        double r3345966 = r3345948 ? r3345958 : r3345965;
        return r3345966;
}

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
Target15.3
Herbie15.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -8.110688788355602e-13 or 1.0915005808183093e-33 < eps

    1. Initial program 30.1

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

    3. Applied tan-sum1.9

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

    5. Applied clear-num2.0

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

    if -8.110688788355602e-13 < eps < 1.0915005808183093e-33

    1. Initial program 45.4

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

    2. Taylor expanded around 0 30.6

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

    3. Simplified30.6

      \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot \left(x \cdot \left(x + \varepsilon\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.110688788355602 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.0915005808183093 \cdot 10^{-33}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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