2sin (example 3.3)

Average Error: 37.4 → 0.7

Time: 22.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -109.7910227234083322400692850351333618164 \lor \neg \left(\varepsilon \le 1.164537554863376176631871181292597059554 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r85029 = x;
        double r85030 = eps;
        double r85031 = r85029 + r85030;
        double r85032 = sin(r85031);
        double r85033 = sin(r85029);
        double r85034 = r85032 - r85033;
        return r85034;
}

↓

double f(double x, double eps) {
        double r85035 = eps;
        double r85036 = -109.79102272340833;
        bool r85037 = r85035 <= r85036;
        double r85038 = 1.1645375548633762e-17;
        bool r85039 = r85035 <= r85038;
        double r85040 = !r85039;
        bool r85041 = r85037 || r85040;
        double r85042 = x;
        double r85043 = sin(r85042);
        double r85044 = cos(r85035);
        double r85045 = r85043 * r85044;
        double r85046 = cos(r85042);
        double r85047 = sin(r85035);
        double r85048 = r85046 * r85047;
        double r85049 = r85045 + r85048;
        double r85050 = r85049 - r85043;
        double r85051 = 2.0;
        double r85052 = r85035 / r85051;
        double r85053 = sin(r85052);
        double r85054 = fma(r85051, r85042, r85035);
        double r85055 = r85054 / r85051;
        double r85056 = cos(r85055);
        double r85057 = r85053 * r85056;
        double r85058 = r85051 * r85057;
        double r85059 = r85041 ? r85050 : r85058;
        return r85059;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.3
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if eps < -109.79102272340833 or 1.1645375548633762e-17 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 0.9

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

   if -109.79102272340833 < eps < 1.1645375548633762e-17

   1. Initial program 45.0

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

   3. Applied diff-sin 45.0

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

   4. Simplified 0.6

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -109.7910227234083322400692850351333618164 \lor \neg \left(\varepsilon \le 1.164537554863376176631871181292597059554 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))