Average Error: 30.4 → 0.6
Time: 21.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{\sqrt{\sin x}} \cdot \frac{1}{\sqrt{\sin x}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \cos x}{\sqrt{\sin x}} \cdot \frac{1}{\sqrt{\sin x}}\\

\end{array}
double f(double x) {
        double r40246 = 1.0;
        double r40247 = x;
        double r40248 = cos(r40247);
        double r40249 = r40246 - r40248;
        double r40250 = sin(r40247);
        double r40251 = r40249 / r40250;
        return r40251;
}
↓
double f(double x) {
        double r40252 = 1.0;
        double r40253 = x;
        double r40254 = cos(r40253);
        double r40255 = r40252 - r40254;
        double r40256 = sin(r40253);
        double r40257 = r40255 / r40256;
        double r40258 = -0.0013085687730050963;
        bool r40259 = r40257 <= r40258;
        double r40260 = exp(r40257);
        double r40261 = log(r40260);
        double r40262 = 4.957178312169811e-08;
        bool r40263 = r40257 <= r40262;
        double r40264 = 0.041666666666666664;
        double r40265 = 3.0;
        double r40266 = pow(r40253, r40265);
        double r40267 = r40264 * r40266;
        double r40268 = 0.004166666666666667;
        double r40269 = 5.0;
        double r40270 = pow(r40253, r40269);
        double r40271 = r40268 * r40270;
        double r40272 = 0.5;
        double r40273 = r40272 * r40253;
        double r40274 = r40271 + r40273;
        double r40275 = r40267 + r40274;
        double r40276 = sqrt(r40256);
        double r40277 = r40255 / r40276;
        double r40278 = 1.0;
        double r40279 = r40278 / r40276;
        double r40280 = r40277 * r40279;
        double r40281 = r40263 ? r40275 : r40280;
        double r40282 = r40259 ? r40261 : r40281;
        return r40282;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0013085687730050963

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    if -0.0013085687730050963 < (/ (- 1.0 (cos x)) (sin x)) < 4.957178312169811e-08

    1. Initial program 60.2

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

    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 4.957178312169811e-08 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.3

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

    3. Applied add-log-exp1.4

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

    5. Applied add-sqr-sqrt1.5

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

    6. Applied *-un-lft-identity1.5

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

    7. Applied times-frac1.6

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

    8. Applied exp-prod1.7

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

    9. Applied log-pow1.5

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

    10. Simplified1.5

      \[\leadsto \frac{1 - \cos x}{\sqrt{\sin x}} \cdot \color{blue}{\frac{1}{\sqrt{\sin x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{\sqrt{\sin x}} \cdot \frac{1}{\sqrt{\sin x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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