2sin (example 3.3)

Average Error: 37.6 → 0.4

Time: 28.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.370205872151545 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1744534610942515 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r13223009 = x;
        double r13223010 = eps;
        double r13223011 = r13223009 + r13223010;
        double r13223012 = sin(r13223011);
        double r13223013 = sin(r13223009);
        double r13223014 = r13223012 - r13223013;
        return r13223014;
}

↓

double f(double x, double eps) {
        double r13223015 = eps;
        double r13223016 = -8.370205872151545e-09;
        bool r13223017 = r13223015 <= r13223016;
        double r13223018 = x;
        double r13223019 = sin(r13223018);
        double r13223020 = cos(r13223015);
        double r13223021 = r13223019 * r13223020;
        double r13223022 = cos(r13223018);
        double r13223023 = sin(r13223015);
        double r13223024 = r13223022 * r13223023;
        double r13223025 = r13223021 + r13223024;
        double r13223026 = r13223025 - r13223019;
        double r13223027 = 1.1744534610942515e-09;
        bool r13223028 = r13223015 <= r13223027;
        double r13223029 = 2.0;
        double r13223030 = r13223015 / r13223029;
        double r13223031 = sin(r13223030);
        double r13223032 = r13223018 + r13223015;
        double r13223033 = r13223032 + r13223018;
        double r13223034 = r13223033 / r13223029;
        double r13223035 = cos(r13223034);
        double r13223036 = r13223031 * r13223035;
        double r13223037 = r13223029 * r13223036;
        double r13223038 = r13223028 ? r13223037 : r13223026;
        double r13223039 = r13223017 ? r13223026 : r13223038;
        return r13223039;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.6
Target	15.2
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 < -8.370205872151545e-09 or 1.1744534610942515e-09 < eps

   1. Initial program 30.4

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

   3. Applied sin-sum 0.6

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

   if -8.370205872151545e-09 < eps < 1.1744534610942515e-09

   1. Initial program 45.2

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

   3. Applied diff-sin 45.2

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

   4. Simplified 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.370205872151545 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1744534610942515 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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