2sin (example 3.3)

Average Error: 36.5 → 0.5

Time: 18.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.131893915803982637690050267343516710028 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.144339914669044167378723460572493433372 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r174024 = x;
        double r174025 = eps;
        double r174026 = r174024 + r174025;
        double r174027 = sin(r174026);
        double r174028 = sin(r174024);
        double r174029 = r174027 - r174028;
        return r174029;
}

↓

double f(double x, double eps) {
        double r174030 = eps;
        double r174031 = -8.131893915803983e-09;
        bool r174032 = r174030 <= r174031;
        double r174033 = 3.144339914669044e-18;
        bool r174034 = r174030 <= r174033;
        double r174035 = !r174034;
        bool r174036 = r174032 || r174035;
        double r174037 = x;
        double r174038 = sin(r174037);
        double r174039 = cos(r174030);
        double r174040 = r174038 * r174039;
        double r174041 = cos(r174037);
        double r174042 = sin(r174030);
        double r174043 = r174041 * r174042;
        double r174044 = r174040 + r174043;
        double r174045 = r174044 - r174038;
        double r174046 = 2.0;
        double r174047 = r174030 / r174046;
        double r174048 = sin(r174047);
        double r174049 = fma(r174046, r174037, r174030);
        double r174050 = r174049 / r174046;
        double r174051 = cos(r174050);
        double r174052 = r174048 * r174051;
        double r174053 = r174046 * r174052;
        double r174054 = r174036 ? r174045 : r174053;
        return r174054;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.5
Target	14.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.131893915803983e-09 or 3.144339914669044e-18 < eps

   1. Initial program 28.9

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

   3. Applied sin-sum 0.8

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

   if -8.131893915803983e-09 < eps < 3.144339914669044e-18

   1. Initial program 44.7

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

   3. Applied diff-sin 44.7

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

   4. Simplified 0.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.131893915803982637690050267343516710028 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.144339914669044167378723460572493433372 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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)))