Average Error: 37.2 → 0.5
Time: 25.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.240915274365558 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2284365724618488 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.240915274365558 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.2284365724618488 \cdot 10^{-08}:\\
\;\;\;\;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 r15962589 = x;
        double r15962590 = eps;
        double r15962591 = r15962589 + r15962590;
        double r15962592 = sin(r15962591);
        double r15962593 = sin(r15962589);
        double r15962594 = r15962592 - r15962593;
        return r15962594;
}
↓
double f(double x, double eps) {
        double r15962595 = eps;
        double r15962596 = -7.240915274365558e-09;
        bool r15962597 = r15962595 <= r15962596;
        double r15962598 = x;
        double r15962599 = sin(r15962598);
        double r15962600 = cos(r15962595);
        double r15962601 = r15962599 * r15962600;
        double r15962602 = cos(r15962598);
        double r15962603 = sin(r15962595);
        double r15962604 = r15962602 * r15962603;
        double r15962605 = r15962601 + r15962604;
        double r15962606 = r15962605 - r15962599;
        double r15962607 = 1.2284365724618488e-08;
        bool r15962608 = r15962595 <= r15962607;
        double r15962609 = 2.0;
        double r15962610 = r15962595 / r15962609;
        double r15962611 = sin(r15962610);
        double r15962612 = r15962598 + r15962595;
        double r15962613 = r15962612 + r15962598;
        double r15962614 = r15962613 / r15962609;
        double r15962615 = cos(r15962614);
        double r15962616 = r15962611 * r15962615;
        double r15962617 = r15962609 * r15962616;
        double r15962618 = r15962608 ? r15962617 : r15962606;
        double r15962619 = r15962597 ? r15962606 : r15962618;
        return r15962619;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target14.5
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 2 regimes

  2. if eps < -7.240915274365558e-09 or 1.2284365724618488e-08 < eps

    1. Initial program 29.6

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -7.240915274365558e-09 < eps < 1.2284365724618488e-08

    1. Initial program 44.8

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

    3. Applied diff-sin44.8

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

    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.240915274365558 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2284365724618488 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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