Average Error: 30.6 → 0.6
Time: 7.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0202867432772082697:\\ \;\;\;\;\log \left(e^{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}}\right)\\ \mathbf{elif}\;x \le 0.026425734950010618:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0202867432772082697:\\
\;\;\;\;\log \left(e^{\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}}\right)\\

\mathbf{elif}\;x \le 0.026425734950010618:\\
\;\;\;\;\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}:\\
\;\;\;\;\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)\\

\end{array}
double f(double x) {
        double r63431 = 1.0;
        double r63432 = x;
        double r63433 = cos(r63432);
        double r63434 = r63431 - r63433;
        double r63435 = sin(r63432);
        double r63436 = r63434 / r63435;
        return r63436;
}
↓
double f(double x) {
        double r63437 = x;
        double r63438 = -0.02028674327720827;
        bool r63439 = r63437 <= r63438;
        double r63440 = 1.0;
        double r63441 = 3.0;
        double r63442 = pow(r63440, r63441);
        double r63443 = cos(r63437);
        double r63444 = pow(r63443, r63441);
        double r63445 = r63442 - r63444;
        double r63446 = r63440 * r63443;
        double r63447 = fma(r63443, r63443, r63446);
        double r63448 = fma(r63440, r63440, r63447);
        double r63449 = sin(r63437);
        double r63450 = r63448 * r63449;
        double r63451 = r63445 / r63450;
        double r63452 = exp(r63451);
        double r63453 = log(r63452);
        double r63454 = 0.026425734950010618;
        bool r63455 = r63437 <= r63454;
        double r63456 = 0.041666666666666664;
        double r63457 = pow(r63437, r63441);
        double r63458 = 0.004166666666666667;
        double r63459 = 5.0;
        double r63460 = pow(r63437, r63459);
        double r63461 = 0.5;
        double r63462 = r63461 * r63437;
        double r63463 = fma(r63458, r63460, r63462);
        double r63464 = fma(r63456, r63457, r63463);
        double r63465 = r63440 / r63449;
        double r63466 = exp(r63465);
        double r63467 = r63443 / r63449;
        double r63468 = exp(r63467);
        double r63469 = r63466 / r63468;
        double r63470 = log(r63469);
        double r63471 = r63455 ? r63464 : r63470;
        double r63472 = r63439 ? r63453 : r63471;
        return r63472;
}

Error

Bits error versus x

Target

Original30.6
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02028674327720827

    1. Initial program 0.8

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

    3. Applied add-log-exp0.9

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

    5. Applied flip3--1.0

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

    6. Applied associate-/l/1.0

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

    7. Simplified1.0

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

    if -0.02028674327720827 < x < 0.026425734950010618

    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 0.026425734950010618 < x

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    5. Applied div-sub1.3

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

    6. Applied exp-diff1.3

      \[\leadsto \log \color{blue}{\left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0202867432772082697:\\ \;\;\;\;\log \left(e^{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}}\right)\\ \mathbf{elif}\;x \le 0.026425734950010618:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +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)))