Average Error: 29.9 → 0.5
Time: 27.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02231606801598658498808624983666959451512:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.02219699502236853819181483515876607270911:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(1 - \cos x\right)\right)} \cdot \frac{1}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02231606801598658498808624983666959451512:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\

\mathbf{elif}\;x \le 0.02219699502236853819181483515876607270911:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r48436 = 1.0;
        double r48437 = x;
        double r48438 = cos(r48437);
        double r48439 = r48436 - r48438;
        double r48440 = sin(r48437);
        double r48441 = r48439 / r48440;
        return r48441;
}
↓
double f(double x) {
        double r48442 = x;
        double r48443 = -0.022316068015986585;
        bool r48444 = r48442 <= r48443;
        double r48445 = 1.0;
        double r48446 = 3.0;
        double r48447 = pow(r48445, r48446);
        double r48448 = cos(r48442);
        double r48449 = pow(r48448, r48446);
        double r48450 = r48447 - r48449;
        double r48451 = sin(r48442);
        double r48452 = r48445 + r48448;
        double r48453 = r48448 * r48452;
        double r48454 = r48445 * r48445;
        double r48455 = r48453 + r48454;
        double r48456 = r48451 * r48455;
        double r48457 = r48450 / r48456;
        double r48458 = 0.022196995022368538;
        bool r48459 = r48442 <= r48458;
        double r48460 = 0.041666666666666664;
        double r48461 = pow(r48442, r48446);
        double r48462 = r48460 * r48461;
        double r48463 = 0.004166666666666667;
        double r48464 = 5.0;
        double r48465 = pow(r48442, r48464);
        double r48466 = r48463 * r48465;
        double r48467 = 0.5;
        double r48468 = r48467 * r48442;
        double r48469 = r48466 + r48468;
        double r48470 = r48462 + r48469;
        double r48471 = exp(1.0);
        double r48472 = r48445 - r48448;
        double r48473 = log(r48472);
        double r48474 = pow(r48471, r48473);
        double r48475 = 1.0;
        double r48476 = r48475 / r48451;
        double r48477 = r48474 * r48476;
        double r48478 = r48459 ? r48470 : r48477;
        double r48479 = r48444 ? r48457 : r48478;
        return r48479;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022316068015986585

    1. Initial program 0.9

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

    3. Applied flip3--1.0

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{\sin x}\]

    4. Applied associate-/l/1.0

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

    5. Simplified1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}\]

    if -0.022316068015986585 < x < 0.022196995022368538

    1. Initial program 59.9

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

    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 0.022196995022368538 < 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

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

    10. Applied div-inv1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02231606801598658498808624983666959451512:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.02219699502236853819181483515876607270911:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(1 - \cos x\right)\right)} \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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