Average Error: 30.1 → 0.5
Time: 7.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0236105927013185481:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.0199371016091038876:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{\frac{\sin x}{1 - \cos x}}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0236105927013185481:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\mathbf{elif}\;x \le 0.0199371016091038876:\\
\;\;\;\;\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}:\\
\;\;\;\;\log \left(e^{\frac{1}{\frac{\sin x}{1 - \cos x}}}\right)\\

\end{array}
double f(double x) {
        double r67425 = 1.0;
        double r67426 = x;
        double r67427 = cos(r67426);
        double r67428 = r67425 - r67427;
        double r67429 = sin(r67426);
        double r67430 = r67428 / r67429;
        return r67430;
}
↓
double f(double x) {
        double r67431 = x;
        double r67432 = -0.023610592701318548;
        bool r67433 = r67431 <= r67432;
        double r67434 = 1.0;
        double r67435 = sin(r67431);
        double r67436 = r67434 / r67435;
        double r67437 = cos(r67431);
        double r67438 = r67437 / r67435;
        double r67439 = r67436 - r67438;
        double r67440 = 0.019937101609103888;
        bool r67441 = r67431 <= r67440;
        double r67442 = 0.041666666666666664;
        double r67443 = 3.0;
        double r67444 = pow(r67431, r67443);
        double r67445 = 0.004166666666666667;
        double r67446 = 5.0;
        double r67447 = pow(r67431, r67446);
        double r67448 = 0.5;
        double r67449 = r67448 * r67431;
        double r67450 = fma(r67445, r67447, r67449);
        double r67451 = fma(r67442, r67444, r67450);
        double r67452 = 1.0;
        double r67453 = r67434 - r67437;
        double r67454 = r67435 / r67453;
        double r67455 = r67452 / r67454;
        double r67456 = exp(r67455);
        double r67457 = log(r67456);
        double r67458 = r67441 ? r67451 : r67457;
        double r67459 = r67433 ? r67439 : r67458;
        return r67459;
}

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 < -0.023610592701318548

    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 -0.023610592701318548 < x < 0.019937101609103888

    1. Initial program 59.7

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

    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 0.019937101609103888 < x

    1. Initial program 0.8

      \[\frac{1 - \cos x}{\sin x}\]
    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 clear-num1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0236105927013185481:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.0199371016091038876:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{\frac{\sin x}{1 - \cos x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))