Average Error: 39.8 → 0.4
Time: 16.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r31370 = x;
        double r31371 = eps;
        double r31372 = r31370 + r31371;
        double r31373 = cos(r31372);
        double r31374 = cos(r31370);
        double r31375 = r31373 - r31374;
        return r31375;
}
↓
double f(double x, double eps) {
        double r31376 = 0.5;
        double r31377 = eps;
        double r31378 = r31376 * r31377;
        double r31379 = sin(r31378);
        double r31380 = x;
        double r31381 = cos(r31380);
        double r31382 = r31379 * r31381;
        double r31383 = cos(r31378);
        double r31384 = sin(r31380);
        double r31385 = r31383 * r31384;
        double r31386 = r31382 + r31385;
        double r31387 = -2.0;
        double r31388 = r31379 * r31387;
        double r31389 = r31386 * r31388;
        return r31389;
}

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-cos34.2

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

  4. Simplified15.5

    \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  5. Taylor expanded around inf 15.5

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

  6. Simplified15.5

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

  8. Applied fma-udef15.5

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

  9. Applied sin-sum0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))