Average Error: 29.9 → 0.9
Time: 7.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0328316610415967866 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 4.742529791284501 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(\sqrt{e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}}\right) + \frac{\frac{1}{\sin x}}{2} \cdot \left(1 - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0328316610415967866 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 4.742529791284501 \cdot 10^{-8}\right):\\
\;\;\;\;\log \left(\sqrt{e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}}\right) + \frac{\frac{1}{\sin x}}{2} \cdot \left(1 - \cos x\right)\\

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

\end{array}
double f(double x) {
        double r82425 = 1.0;
        double r82426 = x;
        double r82427 = cos(r82426);
        double r82428 = r82425 - r82427;
        double r82429 = sin(r82426);
        double r82430 = r82428 / r82429;
        return r82430;
}
↓
double f(double x) {
        double r82431 = 1.0;
        double r82432 = x;
        double r82433 = cos(r82432);
        double r82434 = r82431 - r82433;
        double r82435 = sin(r82432);
        double r82436 = r82434 / r82435;
        double r82437 = -0.03283166104159679;
        bool r82438 = r82436 <= r82437;
        double r82439 = 4.7425297912845007e-08;
        bool r82440 = r82436 <= r82439;
        double r82441 = !r82440;
        bool r82442 = r82438 || r82441;
        double r82443 = log(r82434);
        double r82444 = exp(r82443);
        double r82445 = r82444 / r82435;
        double r82446 = exp(r82445);
        double r82447 = sqrt(r82446);
        double r82448 = log(r82447);
        double r82449 = 1.0;
        double r82450 = r82449 / r82435;
        double r82451 = 2.0;
        double r82452 = r82450 / r82451;
        double r82453 = r82452 * r82434;
        double r82454 = r82448 + r82453;
        double r82455 = 0.041666666666666664;
        double r82456 = 3.0;
        double r82457 = pow(r82432, r82456);
        double r82458 = r82455 * r82457;
        double r82459 = 0.004166666666666667;
        double r82460 = 5.0;
        double r82461 = pow(r82432, r82460);
        double r82462 = r82459 * r82461;
        double r82463 = 0.5;
        double r82464 = r82463 * r82432;
        double r82465 = r82462 + r82464;
        double r82466 = r82458 + r82465;
        double r82467 = r82442 ? r82454 : r82466;
        return r82467;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.03283166104159679 or 4.7425297912845007e-08 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.0

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

    3. Applied add-log-exp1.1

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

    5. Applied add-exp-log1.1

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

    7. Applied add-sqr-sqrt1.4

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

    8. Applied log-prod1.3

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

    10. Applied div-inv1.3

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

    11. Applied exp-prod1.3

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

    12. Applied sqrt-pow11.3

      \[\leadsto \log \left(\sqrt{e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}}\right) + \log \color{blue}{\left({\left(e^{e^{\log \left(1 - \cos x\right)}}\right)}^{\left(\frac{\frac{1}{\sin x}}{2}\right)}\right)}\]

    13. Applied log-pow1.2

      \[\leadsto \log \left(\sqrt{e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}}\right) + \color{blue}{\frac{\frac{1}{\sin x}}{2} \cdot \log \left(e^{e^{\log \left(1 - \cos x\right)}}\right)}\]

    14. Simplified1.2

      \[\leadsto \log \left(\sqrt{e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}}\right) + \frac{\frac{1}{\sin x}}{2} \cdot \color{blue}{\left(1 - \cos x\right)}\]

    if -0.03283166104159679 < (/ (- 1.0 (cos x)) (sin x)) < 4.7425297912845007e-08

    1. Initial program 59.6

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

    2. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0328316610415967866 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 4.742529791284501 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(\sqrt{e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}}\right) + \frac{\frac{1}{\sin x}}{2} \cdot \left(1 - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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