Average Error: 29.6 → 0.5
Time: 7.4s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.022144219019054306:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0203937394598384565:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin x}}{\frac{1}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.022144219019054306:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\

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

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

\end{array}
double f(double x) {
        double r73027 = 1.0;
        double r73028 = x;
        double r73029 = cos(r73028);
        double r73030 = r73027 - r73029;
        double r73031 = sin(r73028);
        double r73032 = r73030 / r73031;
        return r73032;
}
↓
double f(double x) {
        double r73033 = x;
        double r73034 = -0.022144219019054306;
        bool r73035 = r73033 <= r73034;
        double r73036 = 1.0;
        double r73037 = 3.0;
        double r73038 = pow(r73036, r73037);
        double r73039 = cos(r73033);
        double r73040 = pow(r73039, r73037);
        double r73041 = r73038 - r73040;
        double r73042 = r73036 * r73039;
        double r73043 = fma(r73039, r73039, r73042);
        double r73044 = fma(r73036, r73036, r73043);
        double r73045 = sin(r73033);
        double r73046 = r73044 * r73045;
        double r73047 = r73041 / r73046;
        double r73048 = 0.020393739459838457;
        bool r73049 = r73033 <= r73048;
        double r73050 = 0.041666666666666664;
        double r73051 = pow(r73033, r73037);
        double r73052 = 0.004166666666666667;
        double r73053 = 5.0;
        double r73054 = pow(r73033, r73053);
        double r73055 = 0.5;
        double r73056 = r73055 * r73033;
        double r73057 = fma(r73052, r73054, r73056);
        double r73058 = fma(r73050, r73051, r73057);
        double r73059 = 1.0;
        double r73060 = r73059 / r73045;
        double r73061 = r73036 - r73039;
        double r73062 = r73059 / r73061;
        double r73063 = r73060 / r73062;
        double r73064 = r73049 ? r73058 : r73063;
        double r73065 = r73035 ? r73047 : r73064;
        return r73065;
}

Error

Bits error versus x

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

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

    if -0.022144219019054306 < x < 0.020393739459838457

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    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

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

Reproduce

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

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

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