2sin (example 3.3)

Average Error: 36.9 → 0.5

Time: 6.3s

Precision: 64

Math

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

↓

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \cos x \cdot \sin \varepsilon\right)\right)\right)\]

C

double f(double x, double eps) {
        double r95762 = x;
        double r95763 = eps;
        double r95764 = r95762 + r95763;
        double r95765 = sin(r95764);
        double r95766 = sin(r95762);
        double r95767 = r95765 - r95766;
        return r95767;
}

↓

double f(double x, double eps) {
        double r95768 = x;
        double r95769 = sin(r95768);
        double r95770 = eps;
        double r95771 = cos(r95770);
        double r95772 = 1.0;
        double r95773 = r95771 - r95772;
        double r95774 = exp(r95773);
        double r95775 = log(r95774);
        double r95776 = cos(r95768);
        double r95777 = sin(r95770);
        double r95778 = r95776 * r95777;
        double r95779 = fma(r95769, r95775, r95778);
        double r95780 = expm1(r95779);
        double r95781 = log1p(r95780);
        return r95781;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.2
Herbie	0.5

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

Derivation

1. Initial program 36.9

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

3. Applied sin-sum 21.7

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

4. Applied associate--l+ 21.7

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

5. Taylor expanded around inf 21.7

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

6. Simplified 0.4

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

8. Applied log1p-expm1-u 0.5

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)\right)\right)}\]
9. Using strategy rm

10. Applied add-log-exp 0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \cos \varepsilon - \color{blue}{\log \left(e^{1}\right)}, \cos x \cdot \sin \varepsilon\right)\right)\right)\]

11. Applied add-log-exp 0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \color{blue}{\log \left(e^{\cos \varepsilon}\right)} - \log \left(e^{1}\right), \cos x \cdot \sin \varepsilon\right)\right)\right)\]

12. Applied diff-log 0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \color{blue}{\log \left(\frac{e^{\cos \varepsilon}}{e^{1}}\right)}, \cos x \cdot \sin \varepsilon\right)\right)\right)\]

13. Simplified 0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \log \color{blue}{\left(e^{\cos \varepsilon - 1}\right)}, \cos x \cdot \sin \varepsilon\right)\right)\right)\]

14. Final simplification 0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \log \left(e^{\cos \varepsilon - 1}\right), \cos x \cdot \sin \varepsilon\right)\right)\right)\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)))