Average Error: 39.5 → 0.7
Time: 23.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68493806133625336628177219600380198905 \cdot 10^{-8}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 1.036014288510746111270565891770978339537 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.68493806133625336628177219600380198905 \cdot 10^{-8}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \le 1.036014288510746111270565891770978339537 \cdot 10^{-6}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r2624422 = x;
        double r2624423 = eps;
        double r2624424 = r2624422 + r2624423;
        double r2624425 = cos(r2624424);
        double r2624426 = cos(r2624422);
        double r2624427 = r2624425 - r2624426;
        return r2624427;
}
↓
double f(double x, double eps) {
        double r2624428 = eps;
        double r2624429 = -2.6849380613362534e-08;
        bool r2624430 = r2624428 <= r2624429;
        double r2624431 = x;
        double r2624432 = cos(r2624431);
        double r2624433 = cos(r2624428);
        double r2624434 = r2624432 * r2624433;
        double r2624435 = sin(r2624431);
        double r2624436 = sin(r2624428);
        double r2624437 = r2624435 * r2624436;
        double r2624438 = r2624432 + r2624437;
        double r2624439 = r2624434 - r2624438;
        double r2624440 = 1.0360142885107461e-06;
        bool r2624441 = r2624428 <= r2624440;
        double r2624442 = -2.0;
        double r2624443 = 2.0;
        double r2624444 = r2624428 / r2624443;
        double r2624445 = sin(r2624444);
        double r2624446 = r2624431 + r2624428;
        double r2624447 = r2624446 + r2624431;
        double r2624448 = r2624447 / r2624443;
        double r2624449 = sin(r2624448);
        double r2624450 = r2624445 * r2624449;
        double r2624451 = r2624442 * r2624450;
        double r2624452 = r2624434 - r2624437;
        double r2624453 = r2624452 - r2624432;
        double r2624454 = r2624441 ? r2624451 : r2624453;
        double r2624455 = r2624430 ? r2624439 : r2624454;
        return r2624455;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.6849380613362534e-08

    1. Initial program 29.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.0

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

    if -2.6849380613362534e-08 < eps < 1.0360142885107461e-06

    1. Initial program 49.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied diff-cos37.9

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

    4. Simplified0.4

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

    if 1.0360142885107461e-06 < eps

    1. Initial program 30.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68493806133625336628177219600380198905 \cdot 10^{-8}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 1.036014288510746111270565891770978339537 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))