Average Error: 37.5 → 0.5
Time: 20.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\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 r6929463 = x;
        double r6929464 = eps;
        double r6929465 = r6929463 + r6929464;
        double r6929466 = sin(r6929465);
        double r6929467 = sin(r6929463);
        double r6929468 = r6929466 - r6929467;
        return r6929468;
}
↓
double f(double x, double eps) {
        double r6929469 = eps;
        double r6929470 = -1.1998709236678226e-08;
        bool r6929471 = r6929469 <= r6929470;
        double r6929472 = x;
        double r6929473 = cos(r6929472);
        double r6929474 = sin(r6929469);
        double r6929475 = r6929473 * r6929474;
        double r6929476 = sin(r6929472);
        double r6929477 = r6929475 - r6929476;
        double r6929478 = cos(r6929469);
        double r6929479 = r6929476 * r6929478;
        double r6929480 = r6929477 + r6929479;
        double r6929481 = 1.1087986248072222e-08;
        bool r6929482 = r6929469 <= r6929481;
        double r6929483 = 2.0;
        double r6929484 = 0.5;
        double r6929485 = r6929484 * r6929469;
        double r6929486 = sin(r6929485);
        double r6929487 = r6929472 + r6929469;
        double r6929488 = r6929487 + r6929472;
        double r6929489 = r6929488 / r6929483;
        double r6929490 = cos(r6929489);
        double r6929491 = r6929486 * r6929490;
        double r6929492 = r6929483 * r6929491;
        double r6929493 = r6929479 + r6929475;
        double r6929494 = r6929493 - r6929476;
        double r6929495 = r6929482 ? r6929492 : r6929494;
        double r6929496 = r6929471 ? r6929480 : r6929495;
        return r6929496;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target15.6
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 < -1.1998709236678226e-08

    1. Initial program 31.1

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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+0.6

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

    if -1.1998709236678226e-08 < eps < 1.1087986248072222e-08

    1. Initial program 44.4

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

    3. Applied diff-sin44.4

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

    4. Simplified0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

    if 1.1087986248072222e-08 < eps

    1. Initial program 30.9

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

    3. Applied sin-sum0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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