2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 18.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.778095247816633 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.9528216764299977 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r1956559 = x;
        double r1956560 = eps;
        double r1956561 = r1956559 + r1956560;
        double r1956562 = sin(r1956561);
        double r1956563 = sin(r1956559);
        double r1956564 = r1956562 - r1956563;
        return r1956564;
}

↓

double f(double x, double eps) {
        double r1956565 = eps;
        double r1956566 = -8.778095247816633e-09;
        bool r1956567 = r1956565 <= r1956566;
        double r1956568 = x;
        double r1956569 = sin(r1956568);
        double r1956570 = cos(r1956565);
        double r1956571 = r1956569 * r1956570;
        double r1956572 = cos(r1956568);
        double r1956573 = sin(r1956565);
        double r1956574 = r1956572 * r1956573;
        double r1956575 = r1956571 + r1956574;
        double r1956576 = r1956575 - r1956569;
        double r1956577 = 1.9528216764299977e-08;
        bool r1956578 = r1956565 <= r1956577;
        double r1956579 = 2.0;
        double r1956580 = 0.5;
        double r1956581 = r1956580 * r1956565;
        double r1956582 = sin(r1956581);
        double r1956583 = r1956568 + r1956565;
        double r1956584 = r1956583 + r1956568;
        double r1956585 = r1956584 / r1956579;
        double r1956586 = cos(r1956585);
        double r1956587 = r1956582 * r1956586;
        double r1956588 = r1956579 * r1956587;
        double r1956589 = r1956578 ? r1956588 : r1956576;
        double r1956590 = r1956567 ? r1956576 : r1956589;
        return r1956590;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.4
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 < -8.778095247816633e-09 or 1.9528216764299977e-08 < eps

   1. Initial program 30.1

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

   3. Applied sin-sum 0.5

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

   if -8.778095247816633e-09 < eps < 1.9528216764299977e-08

   1. Initial program 44.6

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

   3. Applied diff-sin 44.6

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

   4. Simplified 0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.778095247816633 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.9528216764299977 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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