2sin (example 3.3)

Average Error: 37.0 → 0.3

Time: 22.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r5500462 = x;
        double r5500463 = eps;
        double r5500464 = r5500462 + r5500463;
        double r5500465 = sin(r5500464);
        double r5500466 = sin(r5500462);
        double r5500467 = r5500465 - r5500466;
        return r5500467;
}

↓

double f(double x, double eps) {
        double r5500468 = x;
        double r5500469 = cos(r5500468);
        double r5500470 = eps;
        double r5500471 = 2.0;
        double r5500472 = r5500470 / r5500471;
        double r5500473 = cos(r5500472);
        double r5500474 = r5500469 * r5500473;
        double r5500475 = sin(r5500468);
        double r5500476 = sin(r5500472);
        double r5500477 = r5500475 * r5500476;
        double r5500478 = r5500474 - r5500477;
        double r5500479 = r5500478 * r5500476;
        double r5500480 = r5500479 * r5500471;
        return r5500480;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.0
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 37.0

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

3. Applied diff-sin 37.3

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

4. Simplified 15.0

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

6. Applied log1p-expm1-u 15.1

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\right)\]

7. Simplified 15.0

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)}\right)\right)\]
8. Using strategy rm

9. Applied fma-udef 15.0

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\right)\right)\]

10. Applied cos-sum 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right)\right)\]

11. Taylor expanded around inf 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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