Average Error: 30.6 → 0.6
Time: 7.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0229853120832149115:\\ \;\;\;\;\log \left(e^{\frac{1}{\frac{\sin x}{1 - \cos x}}}\right)\\ \mathbf{elif}\;x \le 0.026425734950010618:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0229853120832149115:\\
\;\;\;\;\log \left(e^{\frac{1}{\frac{\sin x}{1 - \cos x}}}\right)\\

\mathbf{elif}\;x \le 0.026425734950010618:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)\\

\end{array}
double f(double x) {
        double r51977 = 1.0;
        double r51978 = x;
        double r51979 = cos(r51978);
        double r51980 = r51977 - r51979;
        double r51981 = sin(r51978);
        double r51982 = r51980 / r51981;
        return r51982;
}
↓
double f(double x) {
        double r51983 = x;
        double r51984 = -0.02298531208321491;
        bool r51985 = r51983 <= r51984;
        double r51986 = 1.0;
        double r51987 = sin(r51983);
        double r51988 = 1.0;
        double r51989 = cos(r51983);
        double r51990 = r51988 - r51989;
        double r51991 = r51987 / r51990;
        double r51992 = r51986 / r51991;
        double r51993 = exp(r51992);
        double r51994 = log(r51993);
        double r51995 = 0.026425734950010618;
        bool r51996 = r51983 <= r51995;
        double r51997 = 0.041666666666666664;
        double r51998 = 3.0;
        double r51999 = pow(r51983, r51998);
        double r52000 = r51997 * r51999;
        double r52001 = 0.004166666666666667;
        double r52002 = 5.0;
        double r52003 = pow(r51983, r52002);
        double r52004 = r52001 * r52003;
        double r52005 = 0.5;
        double r52006 = r52005 * r51983;
        double r52007 = r52004 + r52006;
        double r52008 = r52000 + r52007;
        double r52009 = r51988 / r51987;
        double r52010 = exp(r52009);
        double r52011 = r51989 / r51987;
        double r52012 = exp(r52011);
        double r52013 = r52010 / r52012;
        double r52014 = log(r52013);
        double r52015 = r51996 ? r52008 : r52014;
        double r52016 = r51985 ? r51994 : r52015;
        return r52016;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.6
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02298531208321491

    1. Initial program 0.8

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied add-log-exp0.9

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
    4. Using strategy rm

    5. Applied clear-num1.0

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

    if -0.02298531208321491 < x < 0.026425734950010618

    1. Initial program 60.0

      \[\frac{1 - \cos x}{\sin x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

    if 0.026425734950010618 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied add-log-exp1.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
    4. Using strategy rm

    5. Applied div-sub1.3

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

    6. Applied exp-diff1.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0229853120832149115:\\ \;\;\;\;\log \left(e^{\frac{1}{\frac{\sin x}{1 - \cos x}}}\right)\\ \mathbf{elif}\;x \le 0.026425734950010618:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))