Average Error: 39.4 → 0.4
Time: 22.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2
double f(double x, double eps) {
        double r1150124 = x;
        double r1150125 = eps;
        double r1150126 = r1150124 + r1150125;
        double r1150127 = cos(r1150126);
        double r1150128 = cos(r1150124);
        double r1150129 = r1150127 - r1150128;
        return r1150129;
}
↓
double f(double x, double eps) {
        double r1150130 = x;
        double r1150131 = cos(r1150130);
        double r1150132 = eps;
        double r1150133 = 0.5;
        double r1150134 = r1150132 * r1150133;
        double r1150135 = sin(r1150134);
        double r1150136 = r1150131 * r1150135;
        double r1150137 = cos(r1150134);
        double r1150138 = sin(r1150130);
        double r1150139 = r1150137 * r1150138;
        double r1150140 = r1150136 + r1150139;
        double r1150141 = 2.0;
        double r1150142 = r1150132 / r1150141;
        double r1150143 = sin(r1150142);
        double r1150144 = r1150140 * r1150143;
        double r1150145 = -2.0;
        double r1150146 = r1150144 * r1150145;
        return r1150146;
}

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. Initial program 39.4

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

  3. Applied diff-cos33.6

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

  4. Simplified14.7

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

  5. Taylor expanded around 0 14.7

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

  7. Applied sin-sum0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))