Average Error: 30.8 → 0.3
Time: 1.2m
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.508948866519329:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x \cdot \cos x}, \frac{\frac{\sin x}{x}}{x}, 1 + \log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \mathsf{fma}\left(\frac{\frac{\sin x}{x}}{x}, \frac{\sin x}{\cos x}, \frac{\sin x}{x}\right)}\right)\right)\\ \mathbf{elif}\;x \le 2.4203399319729506:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{9}{40}, \frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x \cdot \cos x}, \frac{\frac{\sin x}{x}}{x}, 1 + \log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \mathsf{fma}\left(\frac{\frac{\sin x}{x}}{x}, \frac{\sin x}{\cos x}, \frac{\sin x}{x}\right)}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.508948866519329:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x \cdot \cos x}, \frac{\frac{\sin x}{x}}{x}, 1 + \log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \mathsf{fma}\left(\frac{\frac{\sin x}{x}}{x}, \frac{\sin x}{\cos x}, \frac{\sin x}{x}\right)}\right)\right)\\

\mathbf{elif}\;x \le 2.4203399319729506:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{9}{40}, \frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{-1}{2}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x \cdot \cos x}, \frac{\frac{\sin x}{x}}{x}, 1 + \log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \mathsf{fma}\left(\frac{\frac{\sin x}{x}}{x}, \frac{\sin x}{\cos x}, \frac{\sin x}{x}\right)}\right)\right)\\

\end{array}
double f(double x) {
        double r1009388 = x;
        double r1009389 = sin(r1009388);
        double r1009390 = r1009388 - r1009389;
        double r1009391 = tan(r1009388);
        double r1009392 = r1009388 - r1009391;
        double r1009393 = r1009390 / r1009392;
        return r1009393;
}
↓
double f(double x) {
        double r1009394 = x;
        double r1009395 = -2.508948866519329;
        bool r1009396 = r1009394 <= r1009395;
        double r1009397 = sin(r1009394);
        double r1009398 = cos(r1009394);
        double r1009399 = r1009398 * r1009398;
        double r1009400 = r1009397 / r1009399;
        double r1009401 = r1009397 / r1009394;
        double r1009402 = r1009401 / r1009394;
        double r1009403 = 1.0;
        double r1009404 = r1009401 / r1009398;
        double r1009405 = r1009397 / r1009398;
        double r1009406 = fma(r1009402, r1009405, r1009401);
        double r1009407 = r1009404 - r1009406;
        double r1009408 = exp(r1009407);
        double r1009409 = log(r1009408);
        double r1009410 = r1009403 + r1009409;
        double r1009411 = fma(r1009400, r1009402, r1009410);
        double r1009412 = 2.4203399319729506;
        bool r1009413 = r1009394 <= r1009412;
        double r1009414 = r1009394 * r1009394;
        double r1009415 = 0.225;
        double r1009416 = -0.009642857142857142;
        double r1009417 = r1009414 * r1009414;
        double r1009418 = r1009416 * r1009417;
        double r1009419 = fma(r1009414, r1009415, r1009418);
        double r1009420 = -0.5;
        double r1009421 = r1009419 + r1009420;
        double r1009422 = r1009413 ? r1009421 : r1009411;
        double r1009423 = r1009396 ? r1009411 : r1009422;
        return r1009423;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.508948866519329 or 2.4203399319729506 < x

    1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

    3. Simplified0.3

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

    5. Applied add-log-exp0.3

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

    if -2.508948866519329 < x < 2.4203399319729506

    1. Initial program 62.6

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{9}{40}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-27}{2800}\right) + \frac{-1}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.508948866519329:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x \cdot \cos x}, \frac{\frac{\sin x}{x}}{x}, 1 + \log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \mathsf{fma}\left(\frac{\frac{\sin x}{x}}{x}, \frac{\sin x}{\cos x}, \frac{\sin x}{x}\right)}\right)\right)\\ \mathbf{elif}\;x \le 2.4203399319729506:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{9}{40}, \frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x \cdot \cos x}, \frac{\frac{\sin x}{x}}{x}, 1 + \log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \mathsf{fma}\left(\frac{\frac{\sin x}{x}}{x}, \frac{\sin x}{\cos x}, \frac{\sin x}{x}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))