Average Error: 39.5 → 0.4
Time: 24.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right)\]
double f(double x, double eps) {
        double r3372012 = x;
        double r3372013 = eps;
        double r3372014 = r3372012 + r3372013;
        double r3372015 = cos(r3372014);
        double r3372016 = cos(r3372012);
        double r3372017 = r3372015 - r3372016;
        return r3372017;
}
↓
double f(double x, double eps) {
        double r3372018 = -2.0;
        double r3372019 = 0.5;
        double r3372020 = eps;
        double r3372021 = r3372019 * r3372020;
        double r3372022 = sin(r3372021);
        double r3372023 = cos(r3372021);
        double r3372024 = x;
        double r3372025 = sin(r3372024);
        double r3372026 = r3372023 * r3372025;
        double r3372027 = cos(r3372024);
        double r3372028 = r3372022 * r3372027;
        double r3372029 = r3372026 + r3372028;
        double r3372030 = r3372022 * r3372029;
        double r3372031 = r3372018 * r3372030;
        return r3372031;
}


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

Error

Bits error versus x

Bits error versus eps

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

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

  6. Simplified15.2

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

  8. Applied fma-udef15.2

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

  9. Applied sin-sum0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))