Average Error: 31.4 → 0.3
Time: 28.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.607460140836350248605413071345537900925:\\ \;\;\;\;\left(1 + \left(\left(\frac{\frac{\sin x}{x}}{\cos x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x \cdot \cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\\ \mathbf{elif}\;x \le 14.28171286865168099211587104946374893188:\\ \;\;\;\;\log \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{9}{40}, 1\right), e^{\frac{-1}{2}}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot e^{\frac{-1}{2}}, \frac{351}{22400}, -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(\frac{\frac{\sin x}{x}}{\cos x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x \cdot \cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.607460140836350248605413071345537900925:\\
\;\;\;\;\left(1 + \left(\left(\frac{\frac{\sin x}{x}}{\cos x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x \cdot \cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\\

\mathbf{elif}\;x \le 14.28171286865168099211587104946374893188:\\
\;\;\;\;\log \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{9}{40}, 1\right), e^{\frac{-1}{2}}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot e^{\frac{-1}{2}}, \frac{351}{22400}, -1\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 + \left(\left(\frac{\frac{\sin x}{x}}{\cos x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x \cdot \cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\\

\end{array}
double f(double x) {
        double r709398 = x;
        double r709399 = sin(r709398);
        double r709400 = r709398 - r709399;
        double r709401 = tan(r709398);
        double r709402 = r709398 - r709401;
        double r709403 = r709400 / r709402;
        return r709403;
}
↓
double f(double x) {
        double r709404 = x;
        double r709405 = -2.6074601408363502;
        bool r709406 = r709404 <= r709405;
        double r709407 = 1.0;
        double r709408 = sin(r709404);
        double r709409 = r709408 / r709404;
        double r709410 = cos(r709404);
        double r709411 = r709409 / r709410;
        double r709412 = r709409 * r709409;
        double r709413 = r709410 * r709410;
        double r709414 = r709412 / r709413;
        double r709415 = r709411 + r709414;
        double r709416 = r709415 - r709409;
        double r709417 = r709407 + r709416;
        double r709418 = r709412 / r709410;
        double r709419 = r709417 - r709418;
        double r709420 = 14.281712868651681;
        bool r709421 = r709404 <= r709420;
        double r709422 = r709404 * r709404;
        double r709423 = 0.225;
        double r709424 = fma(r709422, r709423, r709407);
        double r709425 = -0.5;
        double r709426 = exp(r709425);
        double r709427 = r709422 * r709422;
        double r709428 = r709427 * r709426;
        double r709429 = 0.015669642857142858;
        double r709430 = -1.0;
        double r709431 = fma(r709428, r709429, r709430);
        double r709432 = fma(r709424, r709426, r709431);
        double r709433 = r709407 + r709432;
        double r709434 = log(r709433);
        double r709435 = r709421 ? r709434 : r709419;
        double r709436 = r709406 ? r709419 : r709435;
        return r709436;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.6074601408363502 or 14.281712868651681 < x

    1. Initial program 0.0

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

    3. Applied log1p-expm1-u0.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - \sin x}{x - \tan x}\right)\right)}\]

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.3

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

    if -2.6074601408363502 < x < 14.281712868651681

    1. Initial program 62.7

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

    3. Applied log1p-expm1-u62.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - \sin x}{x - \tan x}\right)\right)}\]

    4. Taylor expanded around 0 0.3

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\frac{9}{40} \cdot \left({x}^{2} \cdot e^{\frac{-1}{2}}\right) + \left(e^{\frac{-1}{2}} + \frac{351}{22400} \cdot \left({x}^{4} \cdot e^{\frac{-1}{2}}\right)\right)\right) - 1}\right)\]

    5. Simplified0.3

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{9}{40}, 1\right), e^{\frac{-1}{2}}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot e^{\frac{-1}{2}}, \frac{351}{22400}, -1\right)\right)}\right)\]
    6. Using strategy rm

    7. Applied log1p-udef0.3

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{9}{40}, 1\right), e^{\frac{-1}{2}}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot e^{\frac{-1}{2}}, \frac{351}{22400}, -1\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.607460140836350248605413071345537900925:\\ \;\;\;\;\left(1 + \left(\left(\frac{\frac{\sin x}{x}}{\cos x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x \cdot \cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\\ \mathbf{elif}\;x \le 14.28171286865168099211587104946374893188:\\ \;\;\;\;\log \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{9}{40}, 1\right), e^{\frac{-1}{2}}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot e^{\frac{-1}{2}}, \frac{351}{22400}, -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(\frac{\frac{\sin x}{x}}{\cos x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x \cdot \cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))