Average Error: 31.1 → 0.1
Time: 22.3s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02976690073964273139384140165475400863215:\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \mathbf{elif}\;x \le 2.491803263157965186991305017727427184582:\\ \;\;\;\;{x}^{2} \cdot \left(\frac{9}{40} - {x}^{2} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{{\left(\sin x\right)}^{2}}{x \cdot x}}{{\left(\cos x\right)}^{2}}\right)\right) - \frac{\frac{\frac{{\left(\sin x\right)}^{2}}{\cos x}}{x}}{x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02976690073964273139384140165475400863215:\\
\;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\

\mathbf{elif}\;x \le 2.491803263157965186991305017727427184582:\\
\;\;\;\;{x}^{2} \cdot \left(\frac{9}{40} - {x}^{2} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(1 - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{{\left(\sin x\right)}^{2}}{x \cdot x}}{{\left(\cos x\right)}^{2}}\right)\right) - \frac{\frac{\frac{{\left(\sin x\right)}^{2}}{\cos x}}{x}}{x}\\

\end{array}
double f(double x) {
        double r22469 = x;
        double r22470 = sin(r22469);
        double r22471 = r22469 - r22470;
        double r22472 = tan(r22469);
        double r22473 = r22469 - r22472;
        double r22474 = r22471 / r22473;
        return r22474;
}
↓
double f(double x) {
        double r22475 = x;
        double r22476 = -0.02976690073964273;
        bool r22477 = r22475 <= r22476;
        double r22478 = sin(r22475);
        double r22479 = r22475 - r22478;
        double r22480 = cos(r22475);
        double r22481 = r22478 / r22480;
        double r22482 = r22475 - r22481;
        double r22483 = r22479 / r22482;
        double r22484 = 2.491803263157965;
        bool r22485 = r22475 <= r22484;
        double r22486 = 2.0;
        double r22487 = pow(r22475, r22486);
        double r22488 = 0.225;
        double r22489 = 0.009642857142857142;
        double r22490 = r22487 * r22489;
        double r22491 = r22488 - r22490;
        double r22492 = r22487 * r22491;
        double r22493 = 0.5;
        double r22494 = r22492 - r22493;
        double r22495 = 1.0;
        double r22496 = r22478 / r22475;
        double r22497 = r22495 - r22496;
        double r22498 = r22481 / r22475;
        double r22499 = pow(r22478, r22486);
        double r22500 = r22475 * r22475;
        double r22501 = r22499 / r22500;
        double r22502 = pow(r22480, r22486);
        double r22503 = r22501 / r22502;
        double r22504 = r22498 + r22503;
        double r22505 = r22497 + r22504;
        double r22506 = r22499 / r22480;
        double r22507 = r22506 / r22475;
        double r22508 = r22507 / r22475;
        double r22509 = r22505 - r22508;
        double r22510 = r22485 ? r22494 : r22509;
        double r22511 = r22477 ? r22483 : r22510;
        return r22511;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02976690073964273

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.02976690073964273 < x < 2.491803263157965

    1. Initial program 63.0

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 2.491803263157965 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02976690073964273139384140165475400863215:\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \mathbf{elif}\;x \le 2.491803263157965186991305017727427184582:\\ \;\;\;\;{x}^{2} \cdot \left(\frac{9}{40} - {x}^{2} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{{\left(\sin x\right)}^{2}}{x \cdot x}}{{\left(\cos x\right)}^{2}}\right)\right) - \frac{\frac{\frac{{\left(\sin x\right)}^{2}}{\cos x}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))