2sin (example 3.3)

Average Error: 37.2 → 0.3

Time: 20.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r3688063 = x;
        double r3688064 = eps;
        double r3688065 = r3688063 + r3688064;
        double r3688066 = sin(r3688065);
        double r3688067 = sin(r3688063);
        double r3688068 = r3688066 - r3688067;
        return r3688068;
}

↓

double f(double x, double eps) {
        double r3688069 = 2.0;
        double r3688070 = eps;
        double r3688071 = r3688070 / r3688069;
        double r3688072 = sin(r3688071);
        double r3688073 = 0.5;
        double r3688074 = r3688070 * r3688073;
        double r3688075 = cos(r3688074);
        double r3688076 = x;
        double r3688077 = cos(r3688076);
        double r3688078 = sin(r3688074);
        double r3688079 = sin(r3688076);
        double r3688080 = r3688078 * r3688079;
        double r3688081 = -r3688080;
        double r3688082 = fma(r3688075, r3688077, r3688081);
        double r3688083 = r3688072 * r3688082;
        double r3688084 = r3688069 * r3688083;
        return r3688084;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.2
Target	14.9
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 37.2

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

3. Applied diff-sin 37.5

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

4. Simplified 14.9

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

5. Taylor expanded around inf 14.9

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

6. Simplified 14.9

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

8. Applied fma-udef 14.9

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

9. Applied cos-sum 0.3

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

11. Applied fma-neg 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019130 +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)))