Average Error: 37.4 → 0.7
Time: 21.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3657083573837753 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 4.683430512114554 \cdot 10^{-22}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.3657083573837753 \cdot 10^{-08}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 4.683430512114554 \cdot 10^{-22}:\\
\;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\right) \cdot 2\\

\mathbf{else}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\end{array}
double f(double x, double eps) {
        double r3880429 = x;
        double r3880430 = eps;
        double r3880431 = r3880429 + r3880430;
        double r3880432 = sin(r3880431);
        double r3880433 = sin(r3880429);
        double r3880434 = r3880432 - r3880433;
        return r3880434;
}
↓
double f(double x, double eps) {
        double r3880435 = eps;
        double r3880436 = -1.3657083573837753e-08;
        bool r3880437 = r3880435 <= r3880436;
        double r3880438 = x;
        double r3880439 = cos(r3880438);
        double r3880440 = sin(r3880435);
        double r3880441 = r3880439 * r3880440;
        double r3880442 = sin(r3880438);
        double r3880443 = r3880441 - r3880442;
        double r3880444 = cos(r3880435);
        double r3880445 = r3880442 * r3880444;
        double r3880446 = r3880443 + r3880445;
        double r3880447 = 4.683430512114554e-22;
        bool r3880448 = r3880435 <= r3880447;
        double r3880449 = 2.0;
        double r3880450 = r3880435 / r3880449;
        double r3880451 = sin(r3880450);
        double r3880452 = r3880438 + r3880435;
        double r3880453 = r3880438 + r3880452;
        double r3880454 = r3880453 / r3880449;
        double r3880455 = cos(r3880454);
        double r3880456 = r3880455 * r3880455;
        double r3880457 = r3880455 * r3880456;
        double r3880458 = cbrt(r3880457);
        double r3880459 = r3880451 * r3880458;
        double r3880460 = r3880459 * r3880449;
        double r3880461 = r3880448 ? r3880460 : r3880446;
        double r3880462 = r3880437 ? r3880446 : r3880461;
        return r3880462;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.3
Herbie0.7
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.3657083573837753e-08 or 4.683430512114554e-22 < eps

    1. Initial program 30.5

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

    3. Applied sin-sum1.0

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

    4. Applied associate--l+1.0

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

    if -1.3657083573837753e-08 < eps < 4.683430512114554e-22

    1. Initial program 44.9

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

    3. Applied diff-sin44.9

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

    4. Simplified0.2

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

    6. Applied add-cbrt-cube0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3657083573837753 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 4.683430512114554 \cdot 10^{-22}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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