Average Error: 36.8 → 0.4
Time: 14.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.007204743972663667055111869075290087494068:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.393531495967023669552983062766560173884 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.007204743972663667055111869075290087494068:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.393531495967023669552983062766560173884 \cdot 10^{-8}:\\
\;\;\;\;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 r6272596 = x;
        double r6272597 = eps;
        double r6272598 = r6272596 + r6272597;
        double r6272599 = sin(r6272598);
        double r6272600 = sin(r6272596);
        double r6272601 = r6272599 - r6272600;
        return r6272601;
}
↓
double f(double x, double eps) {
        double r6272602 = eps;
        double r6272603 = -0.007204743972663667;
        bool r6272604 = r6272602 <= r6272603;
        double r6272605 = x;
        double r6272606 = sin(r6272605);
        double r6272607 = cos(r6272602);
        double r6272608 = r6272606 * r6272607;
        double r6272609 = cos(r6272605);
        double r6272610 = sin(r6272602);
        double r6272611 = r6272609 * r6272610;
        double r6272612 = r6272608 + r6272611;
        double r6272613 = r6272612 - r6272606;
        double r6272614 = 1.3935314959670237e-08;
        bool r6272615 = r6272602 <= r6272614;
        double r6272616 = 2.0;
        double r6272617 = r6272602 / r6272616;
        double r6272618 = sin(r6272617);
        double r6272619 = r6272605 + r6272602;
        double r6272620 = r6272619 + r6272605;
        double r6272621 = r6272620 / r6272616;
        double r6272622 = cos(r6272621);
        double r6272623 = r6272618 * r6272622;
        double r6272624 = r6272616 * r6272623;
        double r6272625 = r6272615 ? r6272624 : r6272613;
        double r6272626 = r6272604 ? r6272613 : r6272625;
        return r6272626;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.0
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 < -0.007204743972663667 or 1.3935314959670237e-08 < eps

    1. Initial program 29.7

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

    3. Applied sin-sum0.5

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

    if -0.007204743972663667 < eps < 1.3935314959670237e-08

    1. Initial program 44.2

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

    3. Applied diff-sin44.2

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

    4. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.007204743972663667055111869075290087494068:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.393531495967023669552983062766560173884 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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