Average Error: 30.3 → 0.5
Time: 23.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023254737060871069620482387563242809847:\\ \;\;\;\;\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02350362276191214974674359439177351305261:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.023254737060871069620482387563242809847:\\
\;\;\;\;\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.02350362276191214974674359439177351305261:\\
\;\;\;\;\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{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}\right)}\\

\end{array}
double f(double x) {
        double r57245 = 1.0;
        double r57246 = x;
        double r57247 = cos(r57246);
        double r57248 = r57245 - r57247;
        double r57249 = sin(r57246);
        double r57250 = r57248 / r57249;
        return r57250;
}
↓
double f(double x) {
        double r57251 = x;
        double r57252 = -0.02325473706087107;
        bool r57253 = r57251 <= r57252;
        double r57254 = 1.0;
        double r57255 = r57254 * r57254;
        double r57256 = cos(r57251);
        double r57257 = r57256 * r57256;
        double r57258 = r57254 * r57256;
        double r57259 = r57257 + r57258;
        double r57260 = r57255 + r57259;
        double r57261 = sin(r57251);
        double r57262 = r57254 + r57256;
        double r57263 = r57256 * r57262;
        double r57264 = fma(r57254, r57254, r57263);
        double r57265 = r57261 * r57264;
        double r57266 = r57254 - r57256;
        double r57267 = r57265 / r57266;
        double r57268 = r57260 / r57267;
        double r57269 = 0.02350362276191215;
        bool r57270 = r57251 <= r57269;
        double r57271 = 0.041666666666666664;
        double r57272 = 3.0;
        double r57273 = pow(r57251, r57272);
        double r57274 = 0.004166666666666667;
        double r57275 = 5.0;
        double r57276 = pow(r57251, r57275);
        double r57277 = 0.5;
        double r57278 = r57277 * r57251;
        double r57279 = fma(r57274, r57276, r57278);
        double r57280 = fma(r57271, r57273, r57279);
        double r57281 = pow(r57254, r57272);
        double r57282 = pow(r57256, r57272);
        double r57283 = log1p(r57282);
        double r57284 = expm1(r57283);
        double r57285 = r57281 - r57284;
        double r57286 = 2.0;
        double r57287 = pow(r57256, r57286);
        double r57288 = r57255 - r57287;
        double r57289 = r57288 / r57266;
        double r57290 = r57256 * r57289;
        double r57291 = fma(r57254, r57254, r57290);
        double r57292 = r57261 * r57291;
        double r57293 = r57285 / r57292;
        double r57294 = r57270 ? r57280 : r57293;
        double r57295 = r57253 ? r57268 : r57294;
        return r57295;
}

Error

Bits error versus x

Target

Original30.3
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02325473706087107

    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

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

    7. Applied difference-cubes1.0

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

    8. Applied associate-/l*1.0

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

    if -0.02325473706087107 < x < 0.02350362276191215

    1. Initial program 60.0

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

    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 0.02350362276191215 < x

    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

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

    7. Applied expm1-log1p-u1.0

      \[\leadsto \frac{{1}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\]
    8. Using strategy rm

    9. Applied flip-+1.0

      \[\leadsto \frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 - \cos x}}\right)}\]

    10. Simplified1.0

      \[\leadsto \frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{\color{blue}{1 \cdot 1 - {\left(\cos x\right)}^{2}}}{1 - \cos x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.023254737060871069620482387563242809847:\\ \;\;\;\;\frac{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}{\frac{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02350362276191214974674359439177351305261:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))