2sin (example 3.3)

Average Error: 36.9 → 0.6

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.893930927518443 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.8942327691411048 \cdot 10^{-20}:\\ \;\;\;\;\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r5496977 = x;
        double r5496978 = eps;
        double r5496979 = r5496977 + r5496978;
        double r5496980 = sin(r5496979);
        double r5496981 = sin(r5496977);
        double r5496982 = r5496980 - r5496981;
        return r5496982;
}

↓

double f(double x, double eps) {
        double r5496983 = eps;
        double r5496984 = -5.893930927518443e-09;
        bool r5496985 = r5496983 <= r5496984;
        double r5496986 = x;
        double r5496987 = sin(r5496986);
        double r5496988 = cos(r5496983);
        double r5496989 = r5496987 * r5496988;
        double r5496990 = cos(r5496986);
        double r5496991 = sin(r5496983);
        double r5496992 = r5496990 * r5496991;
        double r5496993 = r5496989 + r5496992;
        double r5496994 = r5496993 - r5496987;
        double r5496995 = 1.8942327691411048e-20;
        bool r5496996 = r5496983 <= r5496995;
        double r5496997 = 2.0;
        double r5496998 = fma(r5496997, r5496986, r5496983);
        double r5496999 = r5496998 / r5496997;
        double r5497000 = cos(r5496999);
        double r5497001 = r5496983 / r5496997;
        double r5497002 = sin(r5497001);
        double r5497003 = r5497000 * r5497002;
        double r5497004 = r5497003 * r5496997;
        double r5497005 = r5496996 ? r5497004 : r5496994;
        double r5497006 = r5496985 ? r5496994 : r5497005;
        return r5497006;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.893930927518443e-09 or 1.8942327691411048e-20 < eps

   1. Initial program 30.1

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

   3. Applied sin-sum 1.0

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

   if -5.893930927518443e-09 < eps < 1.8942327691411048e-20

   1. Initial program 44.5

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

   3. Applied diff-sin 44.5

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

   4. Simplified 0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.893930927518443 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.8942327691411048 \cdot 10^{-20}:\\ \;\;\;\;\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))