Average Error: 39.6 → 1.0
Time: 37.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r4604033 = x;
        double r4604034 = eps;
        double r4604035 = r4604033 + r4604034;
        double r4604036 = cos(r4604035);
        double r4604037 = cos(r4604033);
        double r4604038 = r4604036 - r4604037;
        return r4604038;
}
↓
double f(double x, double eps) {
        double r4604039 = 0.5;
        double r4604040 = eps;
        double r4604041 = r4604039 * r4604040;
        double r4604042 = sin(r4604041);
        double r4604043 = x;
        double r4604044 = cos(r4604043);
        double r4604045 = r4604042 * r4604044;
        double r4604046 = sin(r4604043);
        double r4604047 = cos(r4604041);
        double r4604048 = r4604046 * r4604047;
        double r4604049 = r4604048 * r4604048;
        double r4604050 = r4604048 * r4604049;
        double r4604051 = cbrt(r4604050);
        double r4604052 = r4604045 + r4604051;
        double r4604053 = -2.0;
        double r4604054 = r4604042 * r4604053;
        double r4604055 = r4604052 * r4604054;
        return r4604055;
}

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 39.6

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

  3. Applied diff-cos33.9

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

  4. Simplified14.7

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

  5. Taylor expanded around -inf 14.7

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

  6. Simplified14.6

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

  8. Applied sin-sum0.4

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

  10. Applied add-cbrt-cube0.4

    \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin x \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)}} + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  11. Applied add-cbrt-cube1.1

    \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)} + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  12. Applied cbrt-unprod1.0

    \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right)}} + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  13. Simplified1.0

    \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sqrt[3]{\color{blue}{\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)}} + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  14. Final simplification1.0

    \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \sqrt[3]{\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)}\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

Reproduce

herbie shell --seed 2019125 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))