Average Error: 30.1 → 0.5
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0195414650842071873:\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.023267676544555443:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{\sin x}\\ \end{array}\]
\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:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r39417 = 1.0;
        double r39418 = x;
        double r39419 = cos(r39418);
        double r39420 = r39417 - r39419;
        double r39421 = sin(r39418);
        double r39422 = r39420 / r39421;
        return r39422;
}
↓
double f(double x) {
        double r39423 = x;
        double r39424 = -0.019541465084207187;
        bool r39425 = r39423 <= r39424;
        double r39426 = 1.0;
        double r39427 = cos(r39423);
        double r39428 = r39426 - r39427;
        double r39429 = sin(r39423);
        double r39430 = r39428 / r39429;
        double r39431 = 0.023267676544555443;
        bool r39432 = r39423 <= r39431;
        double r39433 = 0.041666666666666664;
        double r39434 = 3.0;
        double r39435 = pow(r39423, r39434);
        double r39436 = r39433 * r39435;
        double r39437 = 0.004166666666666667;
        double r39438 = 5.0;
        double r39439 = pow(r39423, r39438);
        double r39440 = r39437 * r39439;
        double r39441 = 0.5;
        double r39442 = r39441 * r39423;
        double r39443 = r39440 + r39442;
        double r39444 = r39436 + r39443;
        double r39445 = exp(1.0);
        double r39446 = log(r39428);
        double r39447 = pow(r39445, r39446);
        double r39448 = r39447 / r39429;
        double r39449 = r39432 ? r39444 : r39448;
        double r39450 = r39425 ? r39430 : r39449;
        return r39450;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 rem-exp-log0.9

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

    if -0.019541465084207187 < x < 0.023267676544555443

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.0

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

    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

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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