2sin (example 3.3)

Average Error: 37.4 → 0.9

Time: 20.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004 \lor \neg \left(\varepsilon \le 5.173054498508759236112133194996942845024 \cdot 10^{-30}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r105100 = x;
        double r105101 = eps;
        double r105102 = r105100 + r105101;
        double r105103 = sin(r105102);
        double r105104 = sin(r105100);
        double r105105 = r105103 - r105104;
        return r105105;
}

↓

double f(double x, double eps) {
        double r105106 = eps;
        double r105107 = -0.01398185726995361;
        bool r105108 = r105106 <= r105107;
        double r105109 = 5.173054498508759e-30;
        bool r105110 = r105106 <= r105109;
        double r105111 = !r105110;
        bool r105112 = r105108 || r105111;
        double r105113 = x;
        double r105114 = sin(r105113);
        double r105115 = cos(r105106);
        double r105116 = r105114 * r105115;
        double r105117 = cos(r105113);
        double r105118 = sin(r105106);
        double r105119 = r105117 * r105118;
        double r105120 = r105116 + r105119;
        double r105121 = r105120 - r105114;
        double r105122 = 2.0;
        double r105123 = r105106 / r105122;
        double r105124 = sin(r105123);
        double r105125 = r105113 + r105106;
        double r105126 = r105125 + r105113;
        double r105127 = r105126 / r105122;
        double r105128 = cos(r105127);
        double r105129 = r105124 * r105128;
        double r105130 = r105122 * r105129;
        double r105131 = r105112 ? r105121 : r105130;
        return r105131;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.3
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.01398185726995361 or 5.173054498508759e-30 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 1.4

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

   if -0.01398185726995361 < eps < 5.173054498508759e-30

   1. Initial program 45.4

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

   3. Applied diff-sin 45.4

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

   4. Simplified 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 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004 \lor \neg \left(\varepsilon \le 5.173054498508759236112133194996942845024 \cdot 10^{-30}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))