2sin (example 3.3)

Average Error: 36.9 → 0.4

Time: 6.4s

Precision: 64

Math

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

↓

\[\left(\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

C

double f(double x, double eps) {
        double r159244 = x;
        double r159245 = eps;
        double r159246 = r159244 + r159245;
        double r159247 = sin(r159246);
        double r159248 = sin(r159244);
        double r159249 = r159247 - r159248;
        return r159249;
}

↓

double f(double x, double eps) {
        double r159250 = x;
        double r159251 = sin(r159250);
        double r159252 = eps;
        double r159253 = cos(r159252);
        double r159254 = 3.0;
        double r159255 = pow(r159253, r159254);
        double r159256 = 1.0;
        double r159257 = r159255 - r159256;
        double r159258 = r159251 * r159257;
        double r159259 = r159253 * r159253;
        double r159260 = r159253 * r159256;
        double r159261 = r159256 + r159260;
        double r159262 = r159259 + r159261;
        double r159263 = r159258 / r159262;
        double r159264 = cos(r159250);
        double r159265 = sin(r159252);
        double r159266 = r159264 * r159265;
        double r159267 = r159263 + r159266;
        double r159268 = -r159251;
        double r159269 = fma(r159268, r159256, r159251);
        double r159270 = r159267 + r159269;
        return r159270;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.2
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 36.9

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

3. Applied sin-sum 21.6

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

5. Applied add-cube-cbrt 22.2

   \[\leadsto \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}}\]

6. Applied add-sqr-sqrt 43.2

   \[\leadsto \color{blue}{\sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon} \cdot \sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}} - \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}\]

7. Applied prod-diff 43.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}, \sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}, -\sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right)}\]

8. Simplified 21.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)} + \mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right)\]

9. Simplified 0.4

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

11. Applied fma-udef 0.4

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

13. Applied flip3-- 0.4

    \[\leadsto \left(\sin x \cdot \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} + \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

14. Applied associate-*r/ 0.4

    \[\leadsto \left(\color{blue}{\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} + \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

15. Final simplification 0.4

    \[\leadsto \left(\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

Reproduce

herbie shell --seed 2020020 +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)))