Average Error: 30.2 → 0.5
Time: 6.9s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01849382956573876121697530550136434612796:\\ \;\;\;\;1 \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02173189858796714321598209096464415779337:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.01849382956573876121697530550136434612796:\\
\;\;\;\;1 \cdot \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 0.02173189858796714321598209096464415779337:\\
\;\;\;\;1 \cdot \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}:\\
\;\;\;\;1 \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{\sin x}\\

\end{array}
double f(double x) {
        double r54192 = 1.0;
        double r54193 = x;
        double r54194 = cos(r54193);
        double r54195 = r54192 - r54194;
        double r54196 = sin(r54193);
        double r54197 = r54195 / r54196;
        return r54197;
}
↓
double f(double x) {
        double r54198 = x;
        double r54199 = -0.01849382956573876;
        bool r54200 = r54198 <= r54199;
        double r54201 = 1.0;
        double r54202 = 1.0;
        double r54203 = 3.0;
        double r54204 = pow(r54202, r54203);
        double r54205 = cos(r54198);
        double r54206 = pow(r54205, r54203);
        double r54207 = r54204 - r54206;
        double r54208 = r54202 * r54205;
        double r54209 = fma(r54205, r54205, r54208);
        double r54210 = fma(r54202, r54202, r54209);
        double r54211 = sin(r54198);
        double r54212 = r54210 * r54211;
        double r54213 = r54207 / r54212;
        double r54214 = r54201 * r54213;
        double r54215 = 0.021731898587967143;
        bool r54216 = r54198 <= r54215;
        double r54217 = 0.041666666666666664;
        double r54218 = pow(r54198, r54203);
        double r54219 = 0.004166666666666667;
        double r54220 = 5.0;
        double r54221 = pow(r54198, r54220);
        double r54222 = 0.5;
        double r54223 = r54222 * r54198;
        double r54224 = fma(r54219, r54221, r54223);
        double r54225 = fma(r54217, r54218, r54224);
        double r54226 = r54201 * r54225;
        double r54227 = r54202 + r54205;
        double r54228 = r54202 * r54202;
        double r54229 = fma(r54205, r54227, r54228);
        double r54230 = r54207 / r54229;
        double r54231 = r54230 / r54211;
        double r54232 = r54201 * r54231;
        double r54233 = r54216 ? r54226 : r54232;
        double r54234 = r54200 ? r54214 : r54233;
        return r54234;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.01849382956573876

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

      \[\leadsto \frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}\]

    4. Applied *-un-lft-identity0.9

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

    5. Applied times-frac0.9

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1 - \cos x}{\sin x}}\]

    6. Simplified0.9

      \[\leadsto \color{blue}{1} \cdot \frac{1 - \cos x}{\sin x}\]
    7. Using strategy rm

    8. Applied flip3--1.0

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

    9. Applied associate-/l/1.0

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

    10. Simplified1.0

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

    if -0.01849382956573876 < x < 0.021731898587967143

    1. Initial program 59.8

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

    3. Applied *-un-lft-identity59.8

      \[\leadsto \frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}\]

    4. Applied *-un-lft-identity59.8

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

    5. Applied times-frac59.8

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1 - \cos x}{\sin x}}\]

    6. Simplified59.8

      \[\leadsto \color{blue}{1} \cdot \frac{1 - \cos x}{\sin x}\]
    7. Using strategy rm

    8. Applied add-log-exp59.8

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

    9. Taylor expanded around 0 0.0

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

    10. Simplified0.0

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

    if 0.021731898587967143 < x

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

      \[\leadsto \frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}\]

    4. Applied *-un-lft-identity0.9

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

    5. Applied times-frac0.9

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1 - \cos x}{\sin x}}\]

    6. Simplified0.9

      \[\leadsto \color{blue}{1} \cdot \frac{1 - \cos x}{\sin x}\]
    7. Using strategy rm

    8. Applied flip3--1.0

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

    9. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01849382956573876121697530550136434612796:\\ \;\;\;\;1 \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02173189858796714321598209096464415779337:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))