Average Error: 29.9 → 0.6
Time: 10.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02667679834109110484363647231020877370611:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.02283663453927311723368198670414130901918:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02667679834109110484363647231020877370611:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\mathbf{elif}\;x \le 0.02283663453927311723368198670414130901918:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\

\end{array}
double f(double x) {
        double r101426 = 1.0;
        double r101427 = x;
        double r101428 = cos(r101427);
        double r101429 = r101426 - r101428;
        double r101430 = sin(r101427);
        double r101431 = r101429 / r101430;
        return r101431;
}
↓
double f(double x) {
        double r101432 = x;
        double r101433 = -0.026676798341091105;
        bool r101434 = r101432 <= r101433;
        double r101435 = 1.0;
        double r101436 = sin(r101432);
        double r101437 = r101435 / r101436;
        double r101438 = cos(r101432);
        double r101439 = r101438 / r101436;
        double r101440 = r101437 - r101439;
        double r101441 = 0.022836634539273117;
        bool r101442 = r101432 <= r101441;
        double r101443 = 0.041666666666666664;
        double r101444 = 3.0;
        double r101445 = pow(r101432, r101444);
        double r101446 = r101443 * r101445;
        double r101447 = 0.004166666666666667;
        double r101448 = 5.0;
        double r101449 = pow(r101432, r101448);
        double r101450 = r101447 * r101449;
        double r101451 = 0.5;
        double r101452 = r101451 * r101432;
        double r101453 = r101450 + r101452;
        double r101454 = r101446 + r101453;
        double r101455 = pow(r101435, r101444);
        double r101456 = pow(r101438, r101444);
        double r101457 = r101455 - r101456;
        double r101458 = exp(r101457);
        double r101459 = log(r101458);
        double r101460 = 2.0;
        double r101461 = pow(r101438, r101460);
        double r101462 = r101435 * r101435;
        double r101463 = r101461 - r101462;
        double r101464 = r101438 - r101435;
        double r101465 = r101463 / r101464;
        double r101466 = r101438 * r101465;
        double r101467 = r101466 + r101462;
        double r101468 = r101459 / r101467;
        double r101469 = r101468 / r101436;
        double r101470 = r101442 ? r101454 : r101469;
        double r101471 = r101434 ? r101440 : r101470;
        return r101471;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.026676798341091105

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    if -0.026676798341091105 < x < 0.022836634539273117

    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 0.022836634539273117 < x

    1. Initial program 1.0

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

    3. Applied flip3--1.1

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

    4. Simplified1.1

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

    6. Applied add-log-exp1.1

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

    7. Applied add-log-exp1.1

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

    8. Applied diff-log1.2

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

    9. Simplified1.1

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

    11. Applied flip-+1.1

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

    12. Simplified1.1

      \[\leadsto \frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{\color{blue}{{\left(\cos x\right)}^{2} - 1 \cdot 1}}{\cos x - 1} + 1 \cdot 1}}{\sin x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02667679834109110484363647231020877370611:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{elif}\;x \le 0.02283663453927311723368198670414130901918:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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