Average Error: 30.3 → 0.5
Time: 10.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02150701321781052846593418337306502508:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.01958263952305929617159208078192023094743:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)\right)}}{\sin x \cdot {e}^{\left(\log \left(1 \cdot 1 + \sqrt[3]{{\left(\cos x \cdot \left(\cos x + 1\right)\right)}^{3}}\right)\right)}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02150701321781052846593418337306502508:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;x \le 0.01958263952305929617159208078192023094743:\\
\;\;\;\;\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}^{3} - {\left(\cos x\right)}^{3}\right)\right)}}{\sin x \cdot {e}^{\left(\log \left(1 \cdot 1 + \sqrt[3]{{\left(\cos x \cdot \left(\cos x + 1\right)\right)}^{3}}\right)\right)}}\\

\end{array}
double f(double x) {
        double r68614 = 1.0;
        double r68615 = x;
        double r68616 = cos(r68615);
        double r68617 = r68614 - r68616;
        double r68618 = sin(r68615);
        double r68619 = r68617 / r68618;
        return r68619;
}
↓
double f(double x) {
        double r68620 = x;
        double r68621 = -0.02150701321781053;
        bool r68622 = r68620 <= r68621;
        double r68623 = 1.0;
        double r68624 = cos(r68620);
        double r68625 = r68623 - r68624;
        double r68626 = sin(r68620);
        double r68627 = r68625 / r68626;
        double r68628 = exp(r68627);
        double r68629 = log(r68628);
        double r68630 = 0.019582639523059296;
        bool r68631 = r68620 <= r68630;
        double r68632 = 0.041666666666666664;
        double r68633 = 3.0;
        double r68634 = pow(r68620, r68633);
        double r68635 = r68632 * r68634;
        double r68636 = 0.004166666666666667;
        double r68637 = 5.0;
        double r68638 = pow(r68620, r68637);
        double r68639 = r68636 * r68638;
        double r68640 = 0.5;
        double r68641 = r68640 * r68620;
        double r68642 = r68639 + r68641;
        double r68643 = r68635 + r68642;
        double r68644 = exp(1.0);
        double r68645 = pow(r68623, r68633);
        double r68646 = pow(r68624, r68633);
        double r68647 = r68645 - r68646;
        double r68648 = log(r68647);
        double r68649 = pow(r68644, r68648);
        double r68650 = r68623 * r68623;
        double r68651 = r68624 + r68623;
        double r68652 = r68624 * r68651;
        double r68653 = pow(r68652, r68633);
        double r68654 = cbrt(r68653);
        double r68655 = r68650 + r68654;
        double r68656 = log(r68655);
        double r68657 = pow(r68644, r68656);
        double r68658 = r68626 * r68657;
        double r68659 = r68649 / r68658;
        double r68660 = r68631 ? r68643 : r68659;
        double r68661 = r68622 ? r68629 : r68660;
        return r68661;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02150701321781053

    1. Initial program 0.9

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

    3. Applied add-log-exp1.0

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

    if -0.02150701321781053 < x < 0.019582639523059296

    1. Initial program 59.8

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

    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 0.019582639523059296 < 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 flip3--1.1

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

    11. Applied log-div1.1

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

    12. Applied pow-sub1.0

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

    13. Applied associate-/l/1.0

      \[\leadsto \color{blue}{\frac{{e}^{\left(\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)\right)}}{\sin x \cdot {e}^{\left(\log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)\right)}}}\]
    14. Using strategy rm

    15. Applied add-cbrt-cube1.1

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

    16. Simplified1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02150701321781052846593418337306502508:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.01958263952305929617159208078192023094743:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)\right)}}{\sin x \cdot {e}^{\left(\log \left(1 \cdot 1 + \sqrt[3]{{\left(\cos x \cdot \left(\cos x + 1\right)\right)}^{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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