2cos (problem 3.3.5)

Average Error: 39.8 → 0.4

Time: 23.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r1804079 = x;
        double r1804080 = eps;
        double r1804081 = r1804079 + r1804080;
        double r1804082 = cos(r1804081);
        double r1804083 = cos(r1804079);
        double r1804084 = r1804082 - r1804083;
        return r1804084;
}

↓

double f(double x, double eps) {
        double r1804085 = eps;
        double r1804086 = 0.5;
        double r1804087 = r1804085 * r1804086;
        double r1804088 = sin(r1804087);
        double r1804089 = -2.0;
        double r1804090 = r1804088 * r1804089;
        double r1804091 = x;
        double r1804092 = cos(r1804091);
        double r1804093 = sin(r1804091);
        double r1804094 = cos(r1804087);
        double r1804095 = r1804093 * r1804094;
        double r1804096 = fma(r1804092, r1804088, r1804095);
        double r1804097 = r1804090 * r1804096;
        return r1804097;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Initial program 39.8

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

3. Applied diff-cos 34.1

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

4. Simplified 15.1

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

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. Simplified 15.0

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

8. Applied fma-udef 15.0

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

9. Applied sin-sum 0.4

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

10. Simplified 0.4

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

11. Simplified 0.4

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

13. Applied fma-def 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))