2sin (example 3.3)

Average Error: 37.1 → 0.3

Time: 25.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r12578787 = x;
        double r12578788 = eps;
        double r12578789 = r12578787 + r12578788;
        double r12578790 = sin(r12578789);
        double r12578791 = sin(r12578787);
        double r12578792 = r12578790 - r12578791;
        return r12578792;
}

↓

double f(double x, double eps) {
        double r12578793 = 2.0;
        double r12578794 = eps;
        double r12578795 = 0.5;
        double r12578796 = r12578794 * r12578795;
        double r12578797 = cos(r12578796);
        double r12578798 = x;
        double r12578799 = cos(r12578798);
        double r12578800 = sin(r12578796);
        double r12578801 = sin(r12578798);
        double r12578802 = r12578800 * r12578801;
        double r12578803 = -r12578802;
        double r12578804 = fma(r12578797, r12578799, r12578803);
        double r12578805 = r12578794 / r12578793;
        double r12578806 = sin(r12578805);
        double r12578807 = r12578804 * r12578806;
        double r12578808 = r12578793 * r12578807;
        return r12578808;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.5
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 37.4

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

4. Simplified 15.5

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

5. Taylor expanded around -inf 15.5

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

6. Simplified 15.5

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

8. Applied fma-udef 15.5

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

9. Applied cos-sum 0.3

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

11. Applied fma-neg 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019124 +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)))