Average Error: 39.9 → 0.3
Time: 29.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)
double f(double x, double eps) {
        double r7195371 = x;
        double r7195372 = eps;
        double r7195373 = r7195371 + r7195372;
        double r7195374 = cos(r7195373);
        double r7195375 = cos(r7195371);
        double r7195376 = r7195374 - r7195375;
        return r7195376;
}
↓
double f(double x, double eps) {
        double r7195377 = -2.0;
        double r7195378 = 0.5;
        double r7195379 = eps;
        double r7195380 = r7195378 * r7195379;
        double r7195381 = sin(r7195380);
        double r7195382 = r7195377 * r7195381;
        double r7195383 = x;
        double r7195384 = cos(r7195383);
        double r7195385 = r7195379 * r7195378;
        double r7195386 = cos(r7195385);
        double r7195387 = sin(r7195383);
        double r7195388 = r7195386 * r7195387;
        double r7195389 = fma(r7195381, r7195384, r7195388);
        double r7195390 = r7195382 * r7195389;
        return r7195390;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 39.9

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

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

  8. Applied fma-udef15.1

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

  9. Applied sin-sum0.3

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

  10. Simplified0.3

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

  12. Applied fma-def0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))