2sin (example 3.3)

Average Error: 37.2 → 0.4

Time: 11.3s

Precision: 64

Math

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

↓

\[\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon - 1\right) \cdot \sin x\]

C

double f(double x, double eps) {
        double r76910 = x;
        double r76911 = eps;
        double r76912 = r76910 + r76911;
        double r76913 = sin(r76912);
        double r76914 = sin(r76910);
        double r76915 = r76913 - r76914;
        return r76915;
}

↓

double f(double x, double eps) {
        double r76916 = x;
        double r76917 = cos(r76916);
        double r76918 = eps;
        double r76919 = sin(r76918);
        double r76920 = r76917 * r76919;
        double r76921 = cos(r76918);
        double r76922 = 1.0;
        double r76923 = r76921 - r76922;
        double r76924 = sin(r76916);
        double r76925 = r76923 * r76924;
        double r76926 = r76920 + r76925;
        return r76926;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.2
Target	15.0
Herbie	0.4

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

Derivation

1. Initial program 37.2

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

3. Applied sin-sum 22.1

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

5. Applied *-un-lft-identity 22.1

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

6. Applied *-un-lft-identity 22.1

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

7. Applied distribute-lft-out-- 22.1

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

8. Simplified 0.4

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

10. Applied add-log-exp 0.4

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

11. Applied add-log-exp 0.4

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

12. Applied diff-log 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))