2sin (example 3.3)

Average Error: 37.0 → 0.5

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.156573731129723388055868311301635498189 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 1.313003327922032909289180230548081681769 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r88484 = x;
        double r88485 = eps;
        double r88486 = r88484 + r88485;
        double r88487 = sin(r88486);
        double r88488 = sin(r88484);
        double r88489 = r88487 - r88488;
        return r88489;
}

↓

double f(double x, double eps) {
        double r88490 = eps;
        double r88491 = -1.1565737311297234e-08;
        bool r88492 = r88490 <= r88491;
        double r88493 = 1.3130033279220329e-17;
        bool r88494 = r88490 <= r88493;
        double r88495 = !r88494;
        bool r88496 = r88492 || r88495;
        double r88497 = x;
        double r88498 = sin(r88497);
        double r88499 = cos(r88490);
        double r88500 = r88498 * r88499;
        double r88501 = cos(r88497);
        double r88502 = sin(r88490);
        double r88503 = r88501 * r88502;
        double r88504 = r88500 + r88503;
        double r88505 = r88504 - r88498;
        double r88506 = 2.0;
        double r88507 = fma(r88506, r88497, r88490);
        double r88508 = r88507 / r88506;
        double r88509 = cos(r88508);
        double r88510 = r88490 / r88506;
        double r88511 = sin(r88510);
        double r88512 = r88509 * r88511;
        double r88513 = r88506 * r88512;
        double r88514 = r88496 ? r88505 : r88513;
        return r88514;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.0
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 < -1.1565737311297234e-08 or 1.3130033279220329e-17 < eps

   1. Initial program 29.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -1.1565737311297234e-08 < eps < 1.3130033279220329e-17

   1. Initial program 44.9

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

   3. Applied diff-sin 44.9

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

   4. Simplified 0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
   5. Using strategy rm

   6. Applied *-commutative 0.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.156573731129723388055868311301635498189 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 1.313003327922032909289180230548081681769 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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)))