2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 20.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r3177917 = x;
        double r3177918 = eps;
        double r3177919 = r3177917 + r3177918;
        double r3177920 = sin(r3177919);
        double r3177921 = sin(r3177917);
        double r3177922 = r3177920 - r3177921;
        return r3177922;
}

↓

double f(double x, double eps) {
        double r3177923 = eps;
        double r3177924 = -8.778095247816633e-09;
        bool r3177925 = r3177923 <= r3177924;
        double r3177926 = x;
        double r3177927 = sin(r3177926);
        double r3177928 = cos(r3177923);
        double r3177929 = r3177927 * r3177928;
        double r3177930 = cos(r3177926);
        double r3177931 = sin(r3177923);
        double r3177932 = r3177930 * r3177931;
        double r3177933 = r3177929 + r3177932;
        double r3177934 = r3177933 - r3177927;
        double r3177935 = 1.9528216764299977e-08;
        bool r3177936 = r3177923 <= r3177935;
        double r3177937 = 2.0;
        double r3177938 = r3177923 / r3177937;
        double r3177939 = sin(r3177938);
        double r3177940 = fma(r3177937, r3177926, r3177923);
        double r3177941 = r3177940 / r3177937;
        double r3177942 = cos(r3177941);
        double r3177943 = r3177939 * r3177942;
        double r3177944 = r3177943 * r3177937;
        double r3177945 = r3177936 ? r3177944 : r3177934;
        double r3177946 = r3177925 ? r3177934 : r3177945;
        return r3177946;
}

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

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

4. Final simplification 0.4

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

Reproduce

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

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

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