Average Error: 30.1 → 0.5
Time: 11.2s
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 r59334 = 1.0;
        double r59335 = x;
        double r59336 = cos(r59335);
        double r59337 = r59334 - r59336;
        double r59338 = sin(r59335);
        double r59339 = r59337 / r59338;
        return r59339;
}
↓
double f(double x) {
        double r59340 = x;
        double r59341 = -0.019541465084207187;
        bool r59342 = r59340 <= r59341;
        double r59343 = 1.0;
        double r59344 = cos(r59340);
        double r59345 = r59343 - r59344;
        double r59346 = sin(r59340);
        double r59347 = r59345 / r59346;
        double r59348 = 0.023267676544555443;
        bool r59349 = r59340 <= r59348;
        double r59350 = 0.041666666666666664;
        double r59351 = 3.0;
        double r59352 = pow(r59340, r59351);
        double r59353 = r59350 * r59352;
        double r59354 = 0.004166666666666667;
        double r59355 = 5.0;
        double r59356 = pow(r59340, r59355);
        double r59357 = r59354 * r59356;
        double r59358 = 0.5;
        double r59359 = r59358 * r59340;
        double r59360 = r59357 + r59359;
        double r59361 = r59353 + r59360;
        double r59362 = exp(1.0);
        double r59363 = log(r59345);
        double r59364 = pow(r59362, r59363);
        double r59365 = r59364 / r59346;
        double r59366 = r59349 ? r59361 : r59365;
        double r59367 = r59342 ? r59347 : r59366;
        return r59367;
}

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-log-exp0.9

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

    5. Applied rem-log-exp0.9

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

    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)))