Average Error: 30.1 → 0.5
Time: 16.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0195414650842071873:\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.023267676544555443:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0195414650842071873:\\
\;\;\;\;\frac{1 - \cos x}{\sin x}\\

\mathbf{elif}\;x \le 0.023267676544555443:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r50991 = 1.0;
        double r50992 = x;
        double r50993 = cos(r50992);
        double r50994 = r50991 - r50993;
        double r50995 = sin(r50992);
        double r50996 = r50994 / r50995;
        return r50996;
}
↓
double f(double x) {
        double r50997 = x;
        double r50998 = -0.019541465084207187;
        bool r50999 = r50997 <= r50998;
        double r51000 = 1.0;
        double r51001 = cos(r50997);
        double r51002 = r51000 - r51001;
        double r51003 = sin(r50997);
        double r51004 = r51002 / r51003;
        double r51005 = 0.023267676544555443;
        bool r51006 = r50997 <= r51005;
        double r51007 = 0.041666666666666664;
        double r51008 = 3.0;
        double r51009 = pow(r50997, r51008);
        double r51010 = r51007 * r51009;
        double r51011 = 0.004166666666666667;
        double r51012 = 5.0;
        double r51013 = pow(r50997, r51012);
        double r51014 = r51011 * r51013;
        double r51015 = 0.5;
        double r51016 = r51015 * r50997;
        double r51017 = r51014 + r51016;
        double r51018 = r51010 + r51017;
        double r51019 = exp(1.0);
        double r51020 = log(r51002);
        double r51021 = pow(r51019, r51020);
        double r51022 = r51021 / r51003;
        double r51023 = r51006 ? r51018 : r51022;
        double r51024 = r50999 ? r51004 : r51023;
        return r51024;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.019541465084207187

    1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
    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 rem-log-exp0.9

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x}}\]

    if -0.019541465084207187 < x < 0.023267676544555443

    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 0.023267676544555443 < x

    1. Initial program 0.9

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

    3. Applied add-exp-log0.9

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

    5. Applied pow10.9

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

    6. Applied log-pow0.9

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

    7. Applied exp-prod1.0

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

    8. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0195414650842071873:\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.023267676544555443:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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