Average Error: 37.1 → 0.4
Time: 19.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2067536048493133 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.162253177009699 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.2067536048493133 \cdot 10^{-08}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 6.162253177009699 \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}
double f(double x, double eps) {
        double r11263144 = x;
        double r11263145 = eps;
        double r11263146 = r11263144 + r11263145;
        double r11263147 = sin(r11263146);
        double r11263148 = sin(r11263144);
        double r11263149 = r11263147 - r11263148;
        return r11263149;
}
↓
double f(double x, double eps) {
        double r11263150 = eps;
        double r11263151 = -1.2067536048493133e-08;
        bool r11263152 = r11263150 <= r11263151;
        double r11263153 = x;
        double r11263154 = sin(r11263153);
        double r11263155 = cos(r11263150);
        double r11263156 = r11263154 * r11263155;
        double r11263157 = cos(r11263153);
        double r11263158 = sin(r11263150);
        double r11263159 = r11263157 * r11263158;
        double r11263160 = r11263156 + r11263159;
        double r11263161 = r11263160 - r11263154;
        double r11263162 = 6.162253177009699e-09;
        bool r11263163 = r11263150 <= r11263162;
        double r11263164 = 2.0;
        double r11263165 = r11263150 / r11263164;
        double r11263166 = sin(r11263165);
        double r11263167 = r11263153 + r11263150;
        double r11263168 = r11263167 + r11263153;
        double r11263169 = r11263168 / r11263164;
        double r11263170 = cos(r11263169);
        double r11263171 = r11263166 * r11263170;
        double r11263172 = r11263164 * r11263171;
        double r11263173 = r11263163 ? r11263172 : r11263161;
        double r11263174 = r11263152 ? r11263161 : r11263173;
        return r11263174;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.5
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.2067536048493133e-08 or 6.162253177009699e-09 < eps

    1. Initial program 30.1

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

    3. Applied sin-sum0.6

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

    if -1.2067536048493133e-08 < eps < 6.162253177009699e-09

    1. Initial program 44.7

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

    3. Applied diff-sin44.7

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

    4. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2067536048493133 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.162253177009699 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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