2sin (example 3.3)

Average Error: 36.9 → 0.6

Time: 20.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r5134092 = x;
        double r5134093 = eps;
        double r5134094 = r5134092 + r5134093;
        double r5134095 = sin(r5134094);
        double r5134096 = sin(r5134092);
        double r5134097 = r5134095 - r5134096;
        return r5134097;
}

↓

double f(double x, double eps) {
        double r5134098 = eps;
        double r5134099 = -5.893930927518443e-09;
        bool r5134100 = r5134098 <= r5134099;
        double r5134101 = x;
        double r5134102 = sin(r5134101);
        double r5134103 = cos(r5134098);
        double r5134104 = r5134102 * r5134103;
        double r5134105 = cos(r5134101);
        double r5134106 = sin(r5134098);
        double r5134107 = r5134105 * r5134106;
        double r5134108 = r5134104 + r5134107;
        double r5134109 = r5134108 - r5134102;
        double r5134110 = 1.8942327691411048e-20;
        bool r5134111 = r5134098 <= r5134110;
        double r5134112 = 2.0;
        double r5134113 = r5134098 / r5134112;
        double r5134114 = sin(r5134113);
        double r5134115 = r5134101 + r5134101;
        double r5134116 = r5134115 + r5134098;
        double r5134117 = r5134116 / r5134112;
        double r5134118 = cos(r5134117);
        double r5134119 = r5134114 * r5134118;
        double r5134120 = r5134112 * r5134119;
        double r5134121 = r5134111 ? r5134120 : r5134109;
        double r5134122 = r5134100 ? r5134109 : r5134121;
        return r5134122;
}

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

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

4. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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