Average Error: 29.6 → 0.5
Time: 7.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.022144219019054306:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.0203937394598384565:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sin x}}{\frac{1}{1 - \cos x}}\\ \end{array}\]
\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 r51382 = 1.0;
        double r51383 = x;
        double r51384 = cos(r51383);
        double r51385 = r51382 - r51384;
        double r51386 = sin(r51383);
        double r51387 = r51385 / r51386;
        return r51387;
}
↓
double f(double x) {
        double r51388 = x;
        double r51389 = -0.022144219019054306;
        bool r51390 = r51388 <= r51389;
        double r51391 = 1.0;
        double r51392 = 3.0;
        double r51393 = pow(r51391, r51392);
        double r51394 = cos(r51388);
        double r51395 = pow(r51394, r51392);
        double r51396 = r51393 - r51395;
        double r51397 = r51391 * r51394;
        double r51398 = fma(r51394, r51394, r51397);
        double r51399 = fma(r51391, r51391, r51398);
        double r51400 = sin(r51388);
        double r51401 = r51399 * r51400;
        double r51402 = r51396 / r51401;
        double r51403 = 0.020393739459838457;
        bool r51404 = r51388 <= r51403;
        double r51405 = 0.041666666666666664;
        double r51406 = pow(r51388, r51392);
        double r51407 = 0.004166666666666667;
        double r51408 = 5.0;
        double r51409 = pow(r51388, r51408);
        double r51410 = 0.5;
        double r51411 = r51410 * r51388;
        double r51412 = fma(r51407, r51409, r51411);
        double r51413 = fma(r51405, r51406, r51412);
        double r51414 = 1.0;
        double r51415 = r51414 / r51400;
        double r51416 = r51391 - r51394;
        double r51417 = r51414 / r51416;
        double r51418 = r51415 / r51417;
        double r51419 = r51404 ? r51413 : r51418;
        double r51420 = r51390 ? r51402 : r51419;
        return r51420;
}

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)))