Average Error: 37.3 → 0.7
Time: 16.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -33857598964.2829437255859375 \lor \neg \left(\varepsilon \le 7.42085573110588448417594255730102433509 \cdot 10^{-12}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -33857598964.2829437255859375 \lor \neg \left(\varepsilon \le 7.42085573110588448417594255730102433509 \cdot 10^{-12}\right):\\
\;\;\;\;\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r222227 = x;
        double r222228 = eps;
        double r222229 = r222227 + r222228;
        double r222230 = sin(r222229);
        double r222231 = sin(r222227);
        double r222232 = r222230 - r222231;
        return r222232;
}
↓
double f(double x, double eps) {
        double r222233 = eps;
        double r222234 = -33857598964.282944;
        bool r222235 = r222233 <= r222234;
        double r222236 = 7.420855731105884e-12;
        bool r222237 = r222233 <= r222236;
        double r222238 = !r222237;
        bool r222239 = r222235 || r222238;
        double r222240 = cos(r222233);
        double r222241 = x;
        double r222242 = sin(r222241);
        double r222243 = r222240 * r222242;
        double r222244 = cos(r222241);
        double r222245 = sin(r222233);
        double r222246 = r222244 * r222245;
        double r222247 = r222243 + r222246;
        double r222248 = r222247 - r222242;
        double r222249 = 2.0;
        double r222250 = r222233 / r222249;
        double r222251 = sin(r222250);
        double r222252 = r222233 + r222241;
        double r222253 = r222241 + r222252;
        double r222254 = r222253 / r222249;
        double r222255 = cos(r222254);
        double r222256 = r222251 * r222255;
        double r222257 = r222249 * r222256;
        double r222258 = r222239 ? r222248 : r222257;
        return r222258;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.5
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 < -33857598964.282944 or 7.420855731105884e-12 < eps

    1. Initial program 30.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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. Simplified0.6

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

    if -33857598964.282944 < eps < 7.420855731105884e-12

    1. Initial program 44.0

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

    3. Applied diff-sin44.0

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

    4. Simplified0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -33857598964.2829437255859375 \lor \neg \left(\varepsilon \le 7.42085573110588448417594255730102433509 \cdot 10^{-12}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))