Average Error: 39.5 → 0.4
Time: 27.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r6403462 = x;
        double r6403463 = eps;
        double r6403464 = r6403462 + r6403463;
        double r6403465 = cos(r6403464);
        double r6403466 = cos(r6403462);
        double r6403467 = r6403465 - r6403466;
        return r6403467;
}
↓
double f(double x, double eps) {
        double r6403468 = 0.5;
        double r6403469 = eps;
        double r6403470 = r6403468 * r6403469;
        double r6403471 = sin(r6403470);
        double r6403472 = x;
        double r6403473 = cos(r6403472);
        double r6403474 = r6403471 * r6403473;
        double r6403475 = sin(r6403472);
        double r6403476 = cos(r6403470);
        double r6403477 = r6403475 * r6403476;
        double r6403478 = r6403474 + r6403477;
        double r6403479 = -2.0;
        double r6403480 = r6403471 * r6403479;
        double r6403481 = r6403478 * r6403480;
        return r6403481;
}

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-cos34.1

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

  4. Simplified15.2

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

  5. Taylor expanded around -inf 15.2

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

  6. Simplified15.2

    \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x + \varepsilon \cdot \frac{1}{2}\right)}\]
  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))