2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 20.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676 \lor \neg \left(\varepsilon \le 8.837665134522452590648921417169903147482 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r55971 = x;
        double r55972 = eps;
        double r55973 = r55971 + r55972;
        double r55974 = sin(r55973);
        double r55975 = sin(r55971);
        double r55976 = r55974 - r55975;
        return r55976;
}

↓

double f(double x, double eps) {
        double r55977 = eps;
        double r55978 = -0.00815188689103541;
        bool r55979 = r55977 <= r55978;
        double r55980 = 8.837665134522453e-09;
        bool r55981 = r55977 <= r55980;
        double r55982 = !r55981;
        bool r55983 = r55979 || r55982;
        double r55984 = x;
        double r55985 = sin(r55984);
        double r55986 = cos(r55977);
        double r55987 = r55985 * r55986;
        double r55988 = cos(r55984);
        double r55989 = sin(r55977);
        double r55990 = r55988 * r55989;
        double r55991 = r55987 + r55990;
        double r55992 = r55991 - r55985;
        double r55993 = 2.0;
        double r55994 = r55977 / r55993;
        double r55995 = sin(r55994);
        double r55996 = r55984 + r55977;
        double r55997 = r55996 + r55984;
        double r55998 = r55997 / r55993;
        double r55999 = cos(r55998);
        double r56000 = r55995 * r55999;
        double r56001 = r55993 * r56000;
        double r56002 = r55983 ? r55992 : r56001;
        return r56002;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.00815188689103541 or 8.837665134522453e-09 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 0.4

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

   if -0.00815188689103541 < eps < 8.837665134522453e-09

   1. Initial program 44.1

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

   3. Applied diff-sin 44.1

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

   4. Simplified 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676 \lor \neg \left(\varepsilon \le 8.837665134522452590648921417169903147482 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))