Average Error: 39.4 → 0.7
Time: 24.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0008099166487036783:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.4877491964720032 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.0008099166487036783:\\
\;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 1.4877491964720032 \cdot 10^{-05}:\\
\;\;\;\;\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r2460100 = x;
        double r2460101 = eps;
        double r2460102 = r2460100 + r2460101;
        double r2460103 = cos(r2460102);
        double r2460104 = cos(r2460100);
        double r2460105 = r2460103 - r2460104;
        return r2460105;
}
↓
double f(double x, double eps) {
        double r2460106 = eps;
        double r2460107 = -0.0008099166487036783;
        bool r2460108 = r2460106 <= r2460107;
        double r2460109 = cos(r2460106);
        double r2460110 = x;
        double r2460111 = cos(r2460110);
        double r2460112 = r2460109 * r2460111;
        double r2460113 = sin(r2460106);
        double r2460114 = sin(r2460110);
        double r2460115 = r2460113 * r2460114;
        double r2460116 = r2460112 - r2460115;
        double r2460117 = r2460116 - r2460111;
        double r2460118 = 1.4877491964720032e-05;
        bool r2460119 = r2460106 <= r2460118;
        double r2460120 = 0.5;
        double r2460121 = r2460120 * r2460106;
        double r2460122 = sin(r2460121);
        double r2460123 = -2.0;
        double r2460124 = r2460106 + r2460110;
        double r2460125 = r2460110 + r2460124;
        double r2460126 = 2.0;
        double r2460127 = r2460125 / r2460126;
        double r2460128 = sin(r2460127);
        double r2460129 = r2460123 * r2460128;
        double r2460130 = r2460122 * r2460129;
        double r2460131 = r2460119 ? r2460130 : r2460117;
        double r2460132 = r2460108 ? r2460117 : r2460131;
        return r2460132;
}

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 2 regimes

  2. if eps < -0.0008099166487036783 or 1.4877491964720032e-05 < eps

    1. Initial program 29.2

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

    3. Applied +-commutative29.2

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

    4. Applied cos-sum0.9

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

    if -0.0008099166487036783 < eps < 1.4877491964720032e-05

    1. Initial program 49.5

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

    3. Applied diff-cos36.8

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

    4. Simplified0.5

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

    6. Applied *-commutative0.5

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

    7. Applied associate-*r*0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0008099166487036783:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.4877491964720032 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))