Average Error: 30.5 → 0.8
Time: 20.6s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.03628702050037945631144609137663792353123 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 1.622109525504273900412314200758512328093 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.03628702050037945631144609137663792353123 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 1.622109525504273900412314200758512328093 \cdot 10^{-6}\right):\\
\;\;\;\;\frac{1 - \cos x}{\sin x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\

\end{array}
double f(double x) {
        double r53976 = 1.0;
        double r53977 = x;
        double r53978 = cos(r53977);
        double r53979 = r53976 - r53978;
        double r53980 = sin(r53977);
        double r53981 = r53979 / r53980;
        return r53981;
}
↓
double f(double x) {
        double r53982 = 1.0;
        double r53983 = x;
        double r53984 = cos(r53983);
        double r53985 = r53982 - r53984;
        double r53986 = sin(r53983);
        double r53987 = r53985 / r53986;
        double r53988 = -0.036287020500379456;
        bool r53989 = r53987 <= r53988;
        double r53990 = 1.622109525504274e-06;
        bool r53991 = r53987 <= r53990;
        double r53992 = !r53991;
        bool r53993 = r53989 || r53992;
        double r53994 = 0.041666666666666664;
        double r53995 = 3.0;
        double r53996 = pow(r53983, r53995);
        double r53997 = 0.004166666666666667;
        double r53998 = 5.0;
        double r53999 = pow(r53983, r53998);
        double r54000 = 0.5;
        double r54001 = r54000 * r53983;
        double r54002 = fma(r53997, r53999, r54001);
        double r54003 = fma(r53994, r53996, r54002);
        double r54004 = r53993 ? r53987 : r54003;
        return r54004;
}

Error

Bits error versus x

Target

Original30.5
Target0.0
Herbie0.8
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.036287020500379456 or 1.622109525504274e-06 < (/ (- 1.0 (cos x)) (sin 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 *-un-lft-identity0.9

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

    6. Applied *-un-lft-identity0.9

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

    7. Applied log-prod0.9

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

    8. Applied exp-sum0.9

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

    9. Applied times-frac0.9

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

    10. Simplified0.9

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

    11. Simplified0.9

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

    if -0.036287020500379456 < (/ (- 1.0 (cos x)) (sin x)) < 1.622109525504274e-06

    1. Initial program 59.5

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.03628702050037945631144609137663792353123 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 1.622109525504273900412314200758512328093 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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