Average Error: 36.9 → 0.8
Time: 21.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\
\;\;\;\;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 r85612 = x;
        double r85613 = eps;
        double r85614 = r85612 + r85613;
        double r85615 = sin(r85614);
        double r85616 = sin(r85612);
        double r85617 = r85615 - r85616;
        return r85617;
}
↓
double f(double x, double eps) {
        double r85618 = eps;
        double r85619 = -8.505892634593632e-09;
        bool r85620 = r85618 <= r85619;
        double r85621 = x;
        double r85622 = sin(r85621);
        double r85623 = cos(r85618);
        double r85624 = r85622 * r85623;
        double r85625 = cos(r85621);
        double r85626 = sin(r85618);
        double r85627 = r85625 * r85626;
        double r85628 = r85627 - r85622;
        double r85629 = r85624 + r85628;
        double r85630 = 8.55589437692315e-30;
        bool r85631 = r85618 <= r85630;
        double r85632 = 2.0;
        double r85633 = r85618 / r85632;
        double r85634 = sin(r85633);
        double r85635 = r85621 + r85618;
        double r85636 = r85635 + r85621;
        double r85637 = r85636 / r85632;
        double r85638 = cos(r85637);
        double r85639 = r85634 * r85638;
        double r85640 = r85632 * r85639;
        double r85641 = r85624 + r85627;
        double r85642 = r85641 - r85622;
        double r85643 = r85631 ? r85640 : r85642;
        double r85644 = r85620 ? r85629 : r85643;
        return r85644;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target14.8
Herbie0.8
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -8.505892634593632e-09

    1. Initial program 28.9

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    if -8.505892634593632e-09 < eps < 8.55589437692315e-30

    1. Initial program 45.9

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

    3. Applied diff-sin45.9

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

    4. Simplified0.2

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

    if 8.55589437692315e-30 < eps

    1. Initial program 29.3

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

    3. Applied sin-sum2.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.505892634593631943814391258018581254419 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 8.5558943769231499470661433859635760918 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))