2sin (example 3.3)

Average Error: 37.6 → 0.4

Time: 16.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r1779122 = x;
        double r1779123 = eps;
        double r1779124 = r1779122 + r1779123;
        double r1779125 = sin(r1779124);
        double r1779126 = sin(r1779122);
        double r1779127 = r1779125 - r1779126;
        return r1779127;
}

↓

double f(double x, double eps) {
        double r1779128 = eps;
        double r1779129 = -8.370205872151545e-09;
        bool r1779130 = r1779128 <= r1779129;
        double r1779131 = x;
        double r1779132 = sin(r1779131);
        double r1779133 = cos(r1779128);
        double r1779134 = r1779132 * r1779133;
        double r1779135 = cos(r1779131);
        double r1779136 = sin(r1779128);
        double r1779137 = r1779135 * r1779136;
        double r1779138 = r1779134 + r1779137;
        double r1779139 = r1779138 - r1779132;
        double r1779140 = 1.1744534610942515e-09;
        bool r1779141 = r1779128 <= r1779140;
        double r1779142 = 2.0;
        double r1779143 = r1779128 / r1779142;
        double r1779144 = sin(r1779143);
        double r1779145 = r1779131 + r1779128;
        double r1779146 = r1779131 + r1779145;
        double r1779147 = r1779146 / r1779142;
        double r1779148 = cos(r1779147);
        double r1779149 = r1779144 * r1779148;
        double r1779150 = r1779142 * r1779149;
        double r1779151 = r1779141 ? r1779150 : r1779139;
        double r1779152 = r1779130 ? r1779139 : r1779151;
        return r1779152;
}

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

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

4. Final simplification 0.4

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

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)))