Average Error: 39.7 → 0.7
Time: 13.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.40285186539907399 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.68480080325460108 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.40285186539907399 \cdot 10^{-5}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.68480080325460108 \cdot 10^{-5}:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r55490 = x;
        double r55491 = eps;
        double r55492 = r55490 + r55491;
        double r55493 = cos(r55492);
        double r55494 = cos(r55490);
        double r55495 = r55493 - r55494;
        return r55495;
}
↓
double f(double x, double eps) {
        double r55496 = eps;
        double r55497 = -4.402851865399074e-05;
        bool r55498 = r55496 <= r55497;
        double r55499 = x;
        double r55500 = cos(r55499);
        double r55501 = cos(r55496);
        double r55502 = r55500 * r55501;
        double r55503 = sin(r55499);
        double r55504 = sin(r55496);
        double r55505 = r55503 * r55504;
        double r55506 = r55502 - r55505;
        double r55507 = r55506 - r55500;
        double r55508 = 2.684800803254601e-05;
        bool r55509 = r55496 <= r55508;
        double r55510 = -2.0;
        double r55511 = 0.5;
        double r55512 = r55511 * r55496;
        double r55513 = sin(r55512);
        double r55514 = r55510 * r55513;
        double r55515 = r55499 + r55496;
        double r55516 = r55515 + r55499;
        double r55517 = 2.0;
        double r55518 = r55516 / r55517;
        double r55519 = sin(r55518);
        double r55520 = r55514 * r55519;
        double r55521 = fma(r55503, r55504, r55500);
        double r55522 = r55502 - r55521;
        double r55523 = r55509 ? r55520 : r55522;
        double r55524 = r55498 ? r55507 : r55523;
        return r55524;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.402851865399074e-05

    1. Initial program 30.2

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

    3. Applied cos-sum0.9

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

    if -4.402851865399074e-05 < eps < 2.684800803254601e-05

    1. Initial program 49.0

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

    3. Applied diff-cos37.5

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

    4. Simplified0.4

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied associate-*r*0.4

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\varepsilon + 0}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)}\]

    7. Simplified0.4

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

    if 2.684800803254601e-05 < eps

    1. Initial program 31.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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

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

    5. Simplified1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.40285186539907399 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.68480080325460108 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))