Average Error: 30.1 → 0.5
Time: 15.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0195414650842071873:\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.023267676544555443:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \end{array}\]
\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:\\
\;\;\;\;\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{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\

\end{array}
double f(double x) {
        double r138614 = 1.0;
        double r138615 = x;
        double r138616 = cos(r138615);
        double r138617 = r138614 - r138616;
        double r138618 = sin(r138615);
        double r138619 = r138617 / r138618;
        return r138619;
}
↓
double f(double x) {
        double r138620 = x;
        double r138621 = -0.019541465084207187;
        bool r138622 = r138620 <= r138621;
        double r138623 = 1.0;
        double r138624 = cos(r138620);
        double r138625 = r138623 - r138624;
        double r138626 = sin(r138620);
        double r138627 = r138625 / r138626;
        double r138628 = 0.023267676544555443;
        bool r138629 = r138620 <= r138628;
        double r138630 = 0.041666666666666664;
        double r138631 = 3.0;
        double r138632 = pow(r138620, r138631);
        double r138633 = 0.004166666666666667;
        double r138634 = 5.0;
        double r138635 = pow(r138620, r138634);
        double r138636 = 0.5;
        double r138637 = r138636 * r138620;
        double r138638 = fma(r138633, r138635, r138637);
        double r138639 = fma(r138630, r138632, r138638);
        double r138640 = exp(1.0);
        double r138641 = log(r138625);
        double r138642 = pow(r138640, r138641);
        double r138643 = r138642 / r138626;
        double r138644 = r138629 ? r138639 : r138643;
        double r138645 = r138622 ? r138627 : r138644;
        return r138645;
}

Error

Bits error versus x

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-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

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

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

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

    11. Applied log-prod0.9

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

    12. Applied unpow-prod-up0.9

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

    13. Simplified0.9

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

    14. Simplified0.9

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

    if -0.019541465084207187 < x < 0.023267676544555443

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.0

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

    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 0.023267676544555443 < 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

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

Reproduce

herbie shell --seed 2020047 +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)))