Average Error: 30.1 → 0.6
Time: 7.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0242859294884205483:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\\ \mathbf{elif}\;x \le 0.0212898443058771626:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0242859294884205483:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\\

\mathbf{elif}\;x \le 0.0212898443058771626:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r43344 = 1.0;
        double r43345 = x;
        double r43346 = cos(r43345);
        double r43347 = r43344 - r43346;
        double r43348 = sin(r43345);
        double r43349 = r43347 / r43348;
        return r43349;
}
↓
double f(double x) {
        double r43350 = x;
        double r43351 = -0.02428592948842055;
        bool r43352 = r43350 <= r43351;
        double r43353 = 1.0;
        double r43354 = 3.0;
        double r43355 = pow(r43353, r43354);
        double r43356 = cos(r43350);
        double r43357 = pow(r43356, r43354);
        double r43358 = r43355 - r43357;
        double r43359 = sin(r43350);
        double r43360 = r43358 / r43359;
        double r43361 = r43353 * r43353;
        double r43362 = r43356 * r43356;
        double r43363 = r43353 * r43356;
        double r43364 = r43362 + r43363;
        double r43365 = r43361 + r43364;
        double r43366 = r43360 / r43365;
        double r43367 = 0.021289844305877163;
        bool r43368 = r43350 <= r43367;
        double r43369 = 0.041666666666666664;
        double r43370 = pow(r43350, r43354);
        double r43371 = r43369 * r43370;
        double r43372 = 0.004166666666666667;
        double r43373 = 5.0;
        double r43374 = pow(r43350, r43373);
        double r43375 = r43372 * r43374;
        double r43376 = 0.5;
        double r43377 = r43376 * r43350;
        double r43378 = r43375 + r43377;
        double r43379 = r43371 + r43378;
        double r43380 = r43353 - r43356;
        double r43381 = sqrt(r43380);
        double r43382 = log(r43381);
        double r43383 = exp(r43382);
        double r43384 = r43359 / r43381;
        double r43385 = r43383 / r43384;
        double r43386 = r43368 ? r43379 : r43385;
        double r43387 = r43352 ? r43366 : r43386;
        return r43387;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02428592948842055

    1. Initial program 0.9

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

    3. Applied div-inv1.0

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

    5. Applied flip3--1.1

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

    6. Applied associate-*l/1.1

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

    7. Simplified1.0

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

    if -0.02428592948842055 < x < 0.021289844305877163

    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 0.021289844305877163 < 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 add-sqr-sqrt1.2

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

    6. Applied log-prod1.2

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

    7. Applied exp-sum1.2

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

    8. Applied associate-/l*1.2

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

    9. Simplified1.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0242859294884205483:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\\ \mathbf{elif}\;x \le 0.0212898443058771626:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{1 - \cos x}\right)}}{\frac{\sin x}{\sqrt{1 - \cos x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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