Average Error: 39.5 → 0.4
Time: 8.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2
double f(double x, double eps) {
        double r78895 = x;
        double r78896 = eps;
        double r78897 = r78895 + r78896;
        double r78898 = cos(r78897);
        double r78899 = cos(r78895);
        double r78900 = r78898 - r78899;
        return r78900;
}
↓
double f(double x, double eps) {
        double r78901 = x;
        double r78902 = sin(r78901);
        double r78903 = 0.5;
        double r78904 = eps;
        double r78905 = r78903 * r78904;
        double r78906 = cos(r78905);
        double r78907 = r78902 * r78906;
        double r78908 = sin(r78905);
        double r78909 = cos(r78901);
        double r78910 = r78908 * r78909;
        double r78911 = r78907 + r78910;
        double r78912 = r78911 * r78908;
        double r78913 = -2.0;
        double r78914 = r78912 * r78913;
        return r78914;
}

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 39.5

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

  3. Applied diff-cos34.0

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

  4. Simplified15.1

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

  6. Applied pow115.1

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

  7. Applied pow115.1

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

  8. Applied pow-prod-down15.1

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

  9. Simplified15.1

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

  11. Applied distribute-lft-in15.1

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

  12. Applied sin-sum0.4

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

  13. Simplified0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))