Average Error: 30.1 → 0.5
Time: 13.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.020168669770568547:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0257297152168081099:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.020168669770568547:\\
\;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\

\mathbf{elif}\;x \le 0.0257297152168081099:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r47328 = 1.0;
        double r47329 = x;
        double r47330 = cos(r47329);
        double r47331 = r47328 - r47330;
        double r47332 = sin(r47329);
        double r47333 = r47331 / r47332;
        return r47333;
}
↓
double f(double x) {
        double r47334 = x;
        double r47335 = -0.020168669770568547;
        bool r47336 = r47334 <= r47335;
        double r47337 = sin(r47334);
        double r47338 = 1.0;
        double r47339 = cos(r47334);
        double r47340 = r47338 - r47339;
        double r47341 = r47337 * r47340;
        double r47342 = r47337 * r47337;
        double r47343 = r47341 / r47342;
        double r47344 = 0.02572971521680811;
        bool r47345 = r47334 <= r47344;
        double r47346 = 0.041666666666666664;
        double r47347 = 3.0;
        double r47348 = pow(r47334, r47347);
        double r47349 = r47346 * r47348;
        double r47350 = 0.004166666666666667;
        double r47351 = 5.0;
        double r47352 = pow(r47334, r47351);
        double r47353 = r47350 * r47352;
        double r47354 = 0.5;
        double r47355 = r47354 * r47334;
        double r47356 = r47353 + r47355;
        double r47357 = r47349 + r47356;
        double r47358 = r47338 / r47337;
        double r47359 = r47339 / r47337;
        double r47360 = r47358 - r47359;
        double r47361 = r47345 ? r47357 : r47360;
        double r47362 = r47336 ? r47343 : r47361;
        return r47362;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.020168669770568547

    1. Initial program 0.9

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

    3. Applied div-sub1.1

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
    4. Using strategy rm

    5. Applied frac-sub1.0

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

    6. Simplified1.0

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

    if -0.020168669770568547 < x < 0.02572971521680811

    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 0.02572971521680811 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.020168669770568547:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0257297152168081099:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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