2sin (example 3.3)

Average Error: 37.1 → 0.3

Time: 23.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r96785 = x;
        double r96786 = eps;
        double r96787 = r96785 + r96786;
        double r96788 = sin(r96787);
        double r96789 = sin(r96785);
        double r96790 = r96788 - r96789;
        return r96790;
}

↓

double f(double x, double eps) {
        double r96791 = 2.0;
        double r96792 = 0.5;
        double r96793 = eps;
        double r96794 = r96792 * r96793;
        double r96795 = cos(r96794);
        double r96796 = x;
        double r96797 = cos(r96796);
        double r96798 = sin(r96796);
        double r96799 = sin(r96794);
        double r96800 = r96798 * r96799;
        double r96801 = -r96800;
        double r96802 = fma(r96795, r96797, r96801);
        double r96803 = r96802 * r96799;
        double r96804 = r96791 * r96803;
        return r96804;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.2
Herbie	0.3

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

Derivation

1. Initial program 37.1

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

3. Applied diff-sin 37.5

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

4. Simplified 15.3

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

5. Taylor expanded around inf 15.3

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

6. Simplified 15.2

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

8. Applied fma-udef 15.2

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

9. Applied cos-sum 0.3

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

10. Taylor expanded around inf 0.3

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

11. Simplified 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019326 +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)))