Average Error: 39.7 → 0.8
Time: 25.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.00045474474793707185:\\ \;\;\;\;\log_* (1 + (e^{\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \le 0.00045474474793707185:\\
\;\;\;\;\log_* (1 + (e^{\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} - 1)^*)\\

\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 r3171050 = x;
        double r3171051 = eps;
        double r3171052 = r3171050 + r3171051;
        double r3171053 = cos(r3171052);
        double r3171054 = cos(r3171050);
        double r3171055 = r3171053 - r3171054;
        return r3171055;
}
↓
double f(double x, double eps) {
        double r3171056 = eps;
        double r3171057 = -2.082748610758504e-07;
        bool r3171058 = r3171056 <= r3171057;
        double r3171059 = x;
        double r3171060 = cos(r3171059);
        double r3171061 = cos(r3171056);
        double r3171062 = r3171060 * r3171061;
        double r3171063 = sin(r3171059);
        double r3171064 = sin(r3171056);
        double r3171065 = r3171063 * r3171064;
        double r3171066 = r3171060 + r3171065;
        double r3171067 = r3171062 - r3171066;
        double r3171068 = 0.00045474474793707185;
        bool r3171069 = r3171056 <= r3171068;
        double r3171070 = 2.0;
        double r3171071 = r3171056 / r3171070;
        double r3171072 = sin(r3171071);
        double r3171073 = -2.0;
        double r3171074 = r3171059 + r3171056;
        double r3171075 = r3171074 + r3171059;
        double r3171076 = r3171075 / r3171070;
        double r3171077 = sin(r3171076);
        double r3171078 = r3171073 * r3171077;
        double r3171079 = r3171072 * r3171078;
        double r3171080 = expm1(r3171079);
        double r3171081 = log1p(r3171080);
        double r3171082 = r3171062 - r3171065;
        double r3171083 = r3171082 - r3171060;
        double r3171084 = r3171069 ? r3171081 : r3171083;
        double r3171085 = r3171058 ? r3171067 : r3171084;
        return r3171085;
}

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 < -2.082748610758504e-07

    1. Initial program 30.1

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

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

    if -2.082748610758504e-07 < eps < 0.00045474474793707185

    1. Initial program 49.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.6

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

    6. Applied associate-*r*0.6

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

    8. Applied log1p-expm1-u0.6

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

    if 0.00045474474793707185 < eps

    1. Initial program 30.3

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

    3. Applied cos-sum0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.00045474474793707185:\\ \;\;\;\;\log_* (1 + (e^{\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))