Average Error: 30.2 → 0.5
Time: 7.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0242809220681906925976001332401210675016:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.02108811747029423755428467757155885919929:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0242809220681906925976001332401210675016:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\mathbf{elif}\;x \le 0.02108811747029423755428467757155885919929:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\

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

\end{array}
double f(double x) {
        double r64712 = 1.0;
        double r64713 = x;
        double r64714 = cos(r64713);
        double r64715 = r64712 - r64714;
        double r64716 = sin(r64713);
        double r64717 = r64715 / r64716;
        return r64717;
}
↓
double f(double x) {
        double r64718 = x;
        double r64719 = -0.024280922068190693;
        bool r64720 = r64718 <= r64719;
        double r64721 = 1.0;
        double r64722 = sin(r64718);
        double r64723 = r64721 / r64722;
        double r64724 = cos(r64718);
        double r64725 = r64724 / r64722;
        double r64726 = r64723 - r64725;
        double r64727 = 0.021088117470294238;
        bool r64728 = r64718 <= r64727;
        double r64729 = 0.041666666666666664;
        double r64730 = 3.0;
        double r64731 = pow(r64718, r64730);
        double r64732 = 0.004166666666666667;
        double r64733 = 5.0;
        double r64734 = pow(r64718, r64733);
        double r64735 = 0.5;
        double r64736 = r64735 * r64718;
        double r64737 = fma(r64732, r64734, r64736);
        double r64738 = fma(r64729, r64731, r64737);
        double r64739 = r64721 - r64724;
        double r64740 = r64722 * r64739;
        double r64741 = r64722 * r64722;
        double r64742 = r64740 / r64741;
        double r64743 = r64728 ? r64738 : r64742;
        double r64744 = r64720 ? r64726 : r64743;
        return r64744;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.024280922068190693

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    if -0.024280922068190693 < x < 0.021088117470294238

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.021088117470294238 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
    4. Using strategy rm

    5. Applied frac-sub1.1

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

    6. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0242809220681906925976001332401210675016:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.02108811747029423755428467757155885919929:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))