2sin (example 3.3)

Average Error: 37.3 → 0.3

Time: 18.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r2393941 = x;
        double r2393942 = eps;
        double r2393943 = r2393941 + r2393942;
        double r2393944 = sin(r2393943);
        double r2393945 = sin(r2393941);
        double r2393946 = r2393944 - r2393945;
        return r2393946;
}

↓

double f(double x, double eps) {
        double r2393947 = 0.5;
        double r2393948 = eps;
        double r2393949 = r2393947 * r2393948;
        double r2393950 = cos(r2393949);
        double r2393951 = x;
        double r2393952 = cos(r2393951);
        double r2393953 = sin(r2393951);
        double r2393954 = 2.0;
        double r2393955 = r2393948 / r2393954;
        double r2393956 = sin(r2393955);
        double r2393957 = r2393953 * r2393956;
        double r2393958 = -r2393957;
        double r2393959 = fma(r2393950, r2393952, r2393958);
        double r2393960 = r2393959 * r2393956;
        double r2393961 = r2393960 * r2393954;
        return r2393961;
}

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

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

8. Applied fma-udef 15.1

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

9. Applied cos-sum 0.3

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

10. Simplified 0.3

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

12. Applied fma-neg 0.3

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

13. Final simplification 0.3

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

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)))