Average Error: 30.1 → 0.5
Time: 12.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.020168669770568547:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.0257297152168081099:\\ \;\;\;\;\mathsf{fma}\left(0.04166666666666663, {x}^{3}, \mathsf{fma}\left(0.004166666666666624, {x}^{5}, 0.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.020168669770568547:\\
\;\;\;\;\frac{{1}^{3} - \mathsf{log1p}\left(\mathsf{expm1}\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)}\\

\mathbf{elif}\;x \le 0.0257297152168081099:\\
\;\;\;\;\mathsf{fma}\left(0.04166666666666663, {x}^{3}, \mathsf{fma}\left(0.004166666666666624, {x}^{5}, 0.5 \cdot x\right)\right)\\

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

\end{array}
double f(double x) {
        double r61245 = 1.0;
        double r61246 = x;
        double r61247 = cos(r61246);
        double r61248 = r61245 - r61247;
        double r61249 = sin(r61246);
        double r61250 = r61248 / r61249;
        return r61250;
}
↓
double f(double x) {
        double r61251 = x;
        double r61252 = -0.020168669770568547;
        bool r61253 = r61251 <= r61252;
        double r61254 = 1.0;
        double r61255 = 3.0;
        double r61256 = pow(r61254, r61255);
        double r61257 = cos(r61251);
        double r61258 = pow(r61257, r61255);
        double r61259 = expm1(r61258);
        double r61260 = log1p(r61259);
        double r61261 = r61256 - r61260;
        double r61262 = sin(r61251);
        double r61263 = r61254 + r61257;
        double r61264 = r61257 * r61263;
        double r61265 = fma(r61254, r61254, r61264);
        double r61266 = r61262 * r61265;
        double r61267 = r61261 / r61266;
        double r61268 = 0.02572971521680811;
        bool r61269 = r61251 <= r61268;
        double r61270 = 0.04166666666666663;
        double r61271 = pow(r61251, r61255);
        double r61272 = 0.004166666666666624;
        double r61273 = 5.0;
        double r61274 = pow(r61251, r61273);
        double r61275 = 0.5;
        double r61276 = r61275 * r61251;
        double r61277 = fma(r61272, r61274, r61276);
        double r61278 = fma(r61270, r61271, r61277);
        double r61279 = r61254 / r61262;
        double r61280 = r61257 / r61262;
        double r61281 = r61279 - r61280;
        double r61282 = r61269 ? r61278 : r61281;
        double r61283 = r61253 ? r61267 : r61282;
        return r61283;
}

Error

Bits error versus x

Target

Original30.1
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.020168669770568547

    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 log1p-expm1-u1.0

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

    if -0.020168669770568547 < x < 0.02572971521680811

    1. Initial program 59.8

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

    3. Applied flip3--59.8

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

    4. Applied associate-/l/59.8

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

    5. Simplified59.8

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

    6. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.04166666666666663 \cdot {x}^{3} + \left(0.004166666666666624 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.04166666666666663, {x}^{3}, \mathsf{fma}\left(0.004166666666666624, {x}^{5}, 0.5 \cdot x\right)\right)}\]

    if 0.02572971521680811 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.020168669770568547:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{3}\right)\right)}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.0257297152168081099:\\ \;\;\;\;\mathsf{fma}\left(0.04166666666666663, {x}^{3}, \mathsf{fma}\left(0.004166666666666624, {x}^{5}, 0.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))