Average Error: 39.8 → 0.8
Time: 15.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\sqrt[3]{\sin x} \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)
double f(double x, double eps) {
        double r33604 = x;
        double r33605 = eps;
        double r33606 = r33604 + r33605;
        double r33607 = cos(r33606);
        double r33608 = cos(r33604);
        double r33609 = r33607 - r33608;
        return r33609;
}
↓
double f(double x, double eps) {
        double r33610 = -2.0;
        double r33611 = 0.5;
        double r33612 = eps;
        double r33613 = r33611 * r33612;
        double r33614 = sin(r33613);
        double r33615 = r33610 * r33614;
        double r33616 = x;
        double r33617 = cos(r33616);
        double r33618 = r33617 * r33614;
        double r33619 = sin(r33616);
        double r33620 = cbrt(r33619);
        double r33621 = r33620 * r33620;
        double r33622 = cos(r33613);
        double r33623 = r33620 * r33622;
        double r33624 = r33621 * r33623;
        double r33625 = r33618 + r33624;
        double r33626 = r33615 * r33625;
        return r33626;
}

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 39.8

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

  3. Applied diff-cos34.4

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

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

  8. Applied sin-sum0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied add-cube-cbrt0.8

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

  13. Applied associate-*l*0.8

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

  14. Final simplification0.8

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

Reproduce

herbie shell --seed 2019212 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))