Average Error: 37.0 → 0.5
Time: 9.6s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{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 r152639 = x;
        double r152640 = eps;
        double r152641 = r152639 + r152640;
        double r152642 = sin(r152641);
        double r152643 = sin(r152639);
        double r152644 = r152642 - r152643;
        return r152644;
}
↓
double f(double x, double eps) {
        double r152645 = eps;
        double r152646 = -6.6823151206119754e-09;
        bool r152647 = r152645 <= r152646;
        double r152648 = x;
        double r152649 = sin(r152648);
        double r152650 = cos(r152645);
        double r152651 = r152649 * r152650;
        double r152652 = cos(r152648);
        double r152653 = sin(r152645);
        double r152654 = r152652 * r152653;
        double r152655 = r152654 - r152649;
        double r152656 = r152651 + r152655;
        double r152657 = 7.436477518387171e-09;
        bool r152658 = r152645 <= r152657;
        double r152659 = 2.0;
        double r152660 = r152648 + r152645;
        double r152661 = r152660 + r152648;
        double r152662 = r152661 / r152659;
        double r152663 = cos(r152662);
        double r152664 = r152645 / r152659;
        double r152665 = sin(r152664);
        double r152666 = r152663 * r152665;
        double r152667 = r152659 * r152666;
        double r152668 = r152651 + r152654;
        double r152669 = r152668 - r152649;
        double r152670 = r152658 ? r152667 : r152669;
        double r152671 = r152647 ? r152656 : r152670;
        return r152671;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.1
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.6823151206119754e-09

    1. Initial program 30.1

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.7

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

    if -6.6823151206119754e-09 < eps < 7.436477518387171e-09

    1. Initial program 45.0

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

    3. Applied diff-sin45.0

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

    4. Simplified0.3

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

    if 7.436477518387171e-09 < eps

    1. Initial program 29.2

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

    3. Applied sin-sum0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))