Average Error: 30.5 → 0.6
Time: 12.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-385364116457663}{18014398509481984} \lor \neg \left(x \le \frac{5347175448897433}{288230376151711744}\right):\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot x + \frac{{x}^{5}}{240}\right) + \frac{{x}^{3}}{24}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-385364116457663}{18014398509481984} \lor \neg \left(x \le \frac{5347175448897433}{288230376151711744}\right):\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

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

\end{array}
double f(double x) {
        double r250659 = 1.0;
        double r250660 = x;
        double r250661 = cos(r250660);
        double r250662 = r250659 - r250661;
        double r250663 = sin(r250660);
        double r250664 = r250662 / r250663;
        return r250664;
}
↓
double f(double x) {
        double r250665 = x;
        double r250666 = -385364116457663.0;
        double r250667 = 18014398509481984.0;
        double r250668 = r250666 / r250667;
        bool r250669 = r250665 <= r250668;
        double r250670 = 5347175448897433.0;
        double r250671 = 2.8823037615171174e+17;
        double r250672 = r250670 / r250671;
        bool r250673 = r250665 <= r250672;
        double r250674 = !r250673;
        bool r250675 = r250669 || r250674;
        double r250676 = 1.0;
        double r250677 = sin(r250665);
        double r250678 = r250676 / r250677;
        double r250679 = cos(r250665);
        double r250680 = r250679 / r250677;
        double r250681 = r250678 - r250680;
        double r250682 = 1.0;
        double r250683 = 2.0;
        double r250684 = r250682 / r250683;
        double r250685 = r250684 * r250665;
        double r250686 = 5.0;
        double r250687 = pow(r250665, r250686);
        double r250688 = 240.0;
        double r250689 = r250687 / r250688;
        double r250690 = r250685 + r250689;
        double r250691 = 3.0;
        double r250692 = pow(r250665, r250691);
        double r250693 = 24.0;
        double r250694 = r250692 / r250693;
        double r250695 = r250690 + r250694;
        double r250696 = r250675 ? r250681 : r250695;
        return r250696;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.5
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.021392005747781384

    1. Initial program 0.9

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

    3. Applied add-exp-log1.0

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

    5. Applied add-sqr-sqrt1.2

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

    6. Applied associate-/l*1.2

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

    7. Simplified1.2

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

    if -0.021392005747781384 < x < 0.018551741562738398

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{24} \cdot {x}^{3}\right) + \frac{{x}^{5}}{240}}\]

    if 0.018551741562738398 < 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-cbrt-cube1.1

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

    6. Simplified1.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-385364116457663}{18014398509481984} \lor \neg \left(x \le \frac{5347175448897433}{288230376151711744}\right):\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot x + \frac{{x}^{5}}{240}\right) + \frac{{x}^{3}}{24}\\ \end{array}\]

Reproduce

herbie shell --seed 350497007 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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