Average Error: 39.8 → 0.9
Time: 16.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.63544918678768572 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 95617.0838819038909\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.63544918678768572 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 95617.0838819038909\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

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

\end{array}
double f(double x, double eps) {
        double r49078 = x;
        double r49079 = eps;
        double r49080 = r49078 + r49079;
        double r49081 = cos(r49080);
        double r49082 = cos(r49078);
        double r49083 = r49081 - r49082;
        return r49083;
}
↓
double f(double x, double eps) {
        double r49084 = eps;
        double r49085 = -0.0009635449186787686;
        bool r49086 = r49084 <= r49085;
        double r49087 = 95617.08388190389;
        bool r49088 = r49084 <= r49087;
        double r49089 = !r49088;
        bool r49090 = r49086 || r49089;
        double r49091 = x;
        double r49092 = cos(r49091);
        double r49093 = cos(r49084);
        double r49094 = r49092 * r49093;
        double r49095 = sin(r49091);
        double r49096 = sin(r49084);
        double r49097 = r49095 * r49096;
        double r49098 = r49097 + r49092;
        double r49099 = r49094 - r49098;
        double r49100 = -2.0;
        double r49101 = 2.0;
        double r49102 = r49084 / r49101;
        double r49103 = sin(r49102);
        double r49104 = r49100 * r49103;
        double r49105 = r49091 + r49084;
        double r49106 = r49105 + r49091;
        double r49107 = r49106 / r49101;
        double r49108 = sin(r49107);
        double r49109 = r49104 * r49108;
        double r49110 = r49090 ? r49099 : r49109;
        return r49110;
}

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 < -0.0009635449186787686 or 95617.08388190389 < eps

    1. Initial program 30.3

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

    if -0.0009635449186787686 < eps < 95617.08388190389

    1. Initial program 49.5

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

    3. Applied diff-cos38.6

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

    4. Simplified1.0

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

    6. Applied associate-*r*1.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.63544918678768572 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 95617.0838819038909\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))