2sin (example 3.3)

Average Error: 37.1 → 0.3

Time: 20.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r2675064 = x;
        double r2675065 = eps;
        double r2675066 = r2675064 + r2675065;
        double r2675067 = sin(r2675066);
        double r2675068 = sin(r2675064);
        double r2675069 = r2675067 - r2675068;
        return r2675069;
}

↓

double f(double x, double eps) {
        double r2675070 = 2.0;
        double r2675071 = x;
        double r2675072 = sin(r2675071);
        double r2675073 = -r2675072;
        double r2675074 = 0.5;
        double r2675075 = eps;
        double r2675076 = r2675074 * r2675075;
        double r2675077 = sin(r2675076);
        double r2675078 = r2675072 * r2675077;
        double r2675079 = fma(r2675073, r2675077, r2675078);
        double r2675080 = cos(r2675076);
        double r2675081 = cos(r2675071);
        double r2675082 = -r2675078;
        double r2675083 = fma(r2675080, r2675081, r2675082);
        double r2675084 = r2675079 + r2675083;
        double r2675085 = r2675084 * r2675077;
        double r2675086 = r2675070 * r2675085;
        return r2675086;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
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 15.3

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

5. Taylor expanded around -inf 15.3

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

6. Simplified 15.3

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

8. Applied fma-udef 15.3

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

9. Applied cos-sum 0.3

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

11. Applied prod-diff 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019133 +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)))