Average Error: 30.0 → 0.6
Time: 24.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004956373674174259849611878792075003730133:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.001149553159340110540173607311942305386765:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), {x}^{5} \cdot \frac{1}{240}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004956373674174259849611878792075003730133:\\
\;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.001149553159340110540173607311942305386765:\\
\;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), {x}^{5} \cdot \frac{1}{240}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\end{array}
double f(double x) {
        double r4261987 = 1.0;
        double r4261988 = x;
        double r4261989 = cos(r4261988);
        double r4261990 = r4261987 - r4261989;
        double r4261991 = sin(r4261988);
        double r4261992 = r4261990 / r4261991;
        return r4261992;
}
↓
double f(double x) {
        double r4261993 = 1.0;
        double r4261994 = x;
        double r4261995 = cos(r4261994);
        double r4261996 = r4261993 - r4261995;
        double r4261997 = sin(r4261994);
        double r4261998 = r4261996 / r4261997;
        double r4261999 = -0.00495637367417426;
        bool r4262000 = r4261998 <= r4261999;
        double r4262001 = exp(1.0);
        double r4262002 = log(r4261996);
        double r4262003 = pow(r4262001, r4262002);
        double r4262004 = r4262003 / r4261997;
        double r4262005 = 0.0011495531593401105;
        bool r4262006 = r4261998 <= r4262005;
        double r4262007 = 0.041666666666666664;
        double r4262008 = r4261994 * r4262007;
        double r4262009 = 0.5;
        double r4262010 = fma(r4261994, r4262008, r4262009);
        double r4262011 = 5.0;
        double r4262012 = pow(r4261994, r4262011);
        double r4262013 = 0.004166666666666667;
        double r4262014 = r4262012 * r4262013;
        double r4262015 = fma(r4261994, r4262010, r4262014);
        double r4262016 = r4261993 / r4261997;
        double r4262017 = r4261995 / r4261997;
        double r4262018 = r4262016 - r4262017;
        double r4262019 = r4262006 ? r4262015 : r4262018;
        double r4262020 = r4262000 ? r4262004 : r4262019;
        return r4262020;
}

Error

Bits error versus x

Target

Original30.0
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.00495637367417426

    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-prod0.9

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

    8. Simplified0.9

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

    if -0.00495637367417426 < (/ (- 1.0 (cos x)) (sin x)) < 0.0011495531593401105

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.2

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

    if 0.0011495531593401105 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 0.9

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

    3. Applied div-sub1.2

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004956373674174259849611878792075003730133:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.001149553159340110540173607311942305386765:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right), {x}^{5} \cdot \frac{1}{240}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))