Average Error: 30.5 → 0.5
Time: 25.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02294790868899147795456627818566630594432:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02393964535351434871901510348379815695807:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1}\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02294790868899147795456627818566630594432:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;x \le 0.02393964535351434871901510348379815695807:\\
\;\;\;\;\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} - {\left(\cos x\right)}^{3}}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1}\right)}\\

\end{array}
double f(double x) {
        double r57292 = 1.0;
        double r57293 = x;
        double r57294 = cos(r57293);
        double r57295 = r57292 - r57294;
        double r57296 = sin(r57293);
        double r57297 = r57295 / r57296;
        return r57297;
}
↓
double f(double x) {
        double r57298 = x;
        double r57299 = -0.022947908688991478;
        bool r57300 = r57298 <= r57299;
        double r57301 = 1.0;
        double r57302 = cos(r57298);
        double r57303 = r57301 - r57302;
        double r57304 = sin(r57298);
        double r57305 = r57303 / r57304;
        double r57306 = exp(r57305);
        double r57307 = log(r57306);
        double r57308 = 0.02393964535351435;
        bool r57309 = r57298 <= r57308;
        double r57310 = 0.041666666666666664;
        double r57311 = 3.0;
        double r57312 = pow(r57298, r57311);
        double r57313 = 0.004166666666666667;
        double r57314 = 5.0;
        double r57315 = pow(r57298, r57314);
        double r57316 = 0.5;
        double r57317 = r57316 * r57298;
        double r57318 = fma(r57313, r57315, r57317);
        double r57319 = fma(r57310, r57312, r57318);
        double r57320 = pow(r57301, r57311);
        double r57321 = pow(r57302, r57311);
        double r57322 = r57320 - r57321;
        double r57323 = 2.0;
        double r57324 = pow(r57302, r57323);
        double r57325 = r57301 * r57301;
        double r57326 = r57324 - r57325;
        double r57327 = r57302 - r57301;
        double r57328 = r57326 / r57327;
        double r57329 = r57302 * r57328;
        double r57330 = fma(r57301, r57301, r57329);
        double r57331 = r57304 * r57330;
        double r57332 = r57322 / r57331;
        double r57333 = r57309 ? r57319 : r57332;
        double r57334 = r57300 ? r57307 : r57333;
        return r57334;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022947908688991478

    1. Initial program 0.9

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

    3. Applied add-log-exp1.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]

    if -0.022947908688991478 < x < 0.02393964535351435

    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 0.02393964535351435 < 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

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

    7. Applied flip-+1.0

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

    8. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02294790868899147795456627818566630594432:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02393964535351434871901510348379815695807:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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)))