2sin (example 3.3)

Average Error: 37.3 → 0.3

Time: 1.8m

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r3437590 = x;
        double r3437591 = eps;
        double r3437592 = r3437590 + r3437591;
        double r3437593 = sin(r3437592);
        double r3437594 = sin(r3437590);
        double r3437595 = r3437593 - r3437594;
        return r3437595;
}

↓

double f(double x, double eps) {
        double r3437596 = 2.0;
        double r3437597 = eps;
        double r3437598 = 0.5;
        double r3437599 = r3437597 * r3437598;
        double r3437600 = cos(r3437599);
        double r3437601 = x;
        double r3437602 = cos(r3437601);
        double r3437603 = sin(r3437601);
        double r3437604 = sin(r3437599);
        double r3437605 = -r3437604;
        double r3437606 = r3437603 * r3437605;
        double r3437607 = fma(r3437600, r3437602, r3437606);
        double r3437608 = r3437607 * r3437604;
        double r3437609 = r3437596 * r3437608;
        return r3437609;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.1
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 37.3

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

3. Applied diff-sin 37.6

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

4. Simplified 15.2

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

5. Taylor expanded around inf 15.2

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

6. Simplified 15.1

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

8. Applied fma-udef 15.1

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

9. Applied cos-sum 0.3

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

11. Applied fma-neg 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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