Average Error: 29.6 → 0.5
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.022144219019054306:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0203937394598384565:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin x}}{\frac{1}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.022144219019054306:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\

\mathbf{elif}\;x \le 0.0203937394598384565:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r40849 = 1.0;
        double r40850 = x;
        double r40851 = cos(r40850);
        double r40852 = r40849 - r40851;
        double r40853 = sin(r40850);
        double r40854 = r40852 / r40853;
        return r40854;
}
↓
double f(double x) {
        double r40855 = x;
        double r40856 = -0.022144219019054306;
        bool r40857 = r40855 <= r40856;
        double r40858 = 1.0;
        double r40859 = 3.0;
        double r40860 = pow(r40858, r40859);
        double r40861 = cos(r40855);
        double r40862 = pow(r40861, r40859);
        double r40863 = r40860 - r40862;
        double r40864 = r40861 + r40858;
        double r40865 = r40861 * r40864;
        double r40866 = r40858 * r40858;
        double r40867 = r40865 + r40866;
        double r40868 = sin(r40855);
        double r40869 = r40867 * r40868;
        double r40870 = r40863 / r40869;
        double r40871 = 0.020393739459838457;
        bool r40872 = r40855 <= r40871;
        double r40873 = 0.041666666666666664;
        double r40874 = pow(r40855, r40859);
        double r40875 = r40873 * r40874;
        double r40876 = 0.004166666666666667;
        double r40877 = 5.0;
        double r40878 = pow(r40855, r40877);
        double r40879 = r40876 * r40878;
        double r40880 = 0.5;
        double r40881 = r40880 * r40855;
        double r40882 = r40879 + r40881;
        double r40883 = r40875 + r40882;
        double r40884 = 1.0;
        double r40885 = r40884 / r40868;
        double r40886 = r40858 - r40861;
        double r40887 = r40884 / r40886;
        double r40888 = r40885 / r40887;
        double r40889 = r40872 ? r40883 : r40888;
        double r40890 = r40857 ? r40870 : r40889;
        return r40890;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022144219019054306

    1. Initial program 0.9

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

    3. Applied flip3--1.0

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

    4. Applied associate-/l/1.0

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

    5. Simplified1.0

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

    if -0.022144219019054306 < x < 0.020393739459838457

    1. Initial program 59.9

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

    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 0.020393739459838457 < x

    1. Initial program 0.9

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

    3. Applied clear-num1.0

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

    5. Applied div-inv1.0

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

    6. Applied associate-/r*1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.022144219019054306:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0203937394598384565:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin x}}{\frac{1}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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