Average Error: 37.1 → 0.7
Time: 18.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.755630184432992 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 7.800526234462867 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.755630184432992 \cdot 10^{-08}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 7.800526234462867 \cdot 10^{-25}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{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 r4013477 = x;
        double r4013478 = eps;
        double r4013479 = r4013477 + r4013478;
        double r4013480 = sin(r4013479);
        double r4013481 = sin(r4013477);
        double r4013482 = r4013480 - r4013481;
        return r4013482;
}
↓
double f(double x, double eps) {
        double r4013483 = eps;
        double r4013484 = -1.755630184432992e-08;
        bool r4013485 = r4013483 <= r4013484;
        double r4013486 = x;
        double r4013487 = sin(r4013486);
        double r4013488 = cos(r4013483);
        double r4013489 = r4013487 * r4013488;
        double r4013490 = cos(r4013486);
        double r4013491 = sin(r4013483);
        double r4013492 = r4013490 * r4013491;
        double r4013493 = r4013489 + r4013492;
        double r4013494 = r4013493 - r4013487;
        double r4013495 = 7.800526234462867e-25;
        bool r4013496 = r4013483 <= r4013495;
        double r4013497 = 2.0;
        double r4013498 = r4013483 / r4013497;
        double r4013499 = sin(r4013498);
        double r4013500 = r4013486 + r4013483;
        double r4013501 = r4013486 + r4013500;
        double r4013502 = r4013501 / r4013497;
        double r4013503 = cos(r4013502);
        double r4013504 = r4013499 * r4013503;
        double r4013505 = r4013497 * r4013504;
        double r4013506 = r4013496 ? r4013505 : r4013494;
        double r4013507 = r4013485 ? r4013494 : r4013506;
        return r4013507;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
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 < -1.755630184432992e-08 or 7.800526234462867e-25 < eps

    1. Initial program 29.9

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

    3. Applied sin-sum1.1

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

    if -1.755630184432992e-08 < eps < 7.800526234462867e-25

    1. Initial program 45.1

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

    3. Applied diff-sin45.1

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

    4. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.755630184432992 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 7.800526234462867 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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