Average Error: 37.0 → 0.6
Time: 20.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.918151501731851694268208750014659269567 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 3.630608855951668331104028504087618457981 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.918151501731851694268208750014659269567 \cdot 10^{-9}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 3.630608855951668331104028504087618457981 \cdot 10^{-18}:\\
\;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\\

\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 r5569235 = x;
        double r5569236 = eps;
        double r5569237 = r5569235 + r5569236;
        double r5569238 = sin(r5569237);
        double r5569239 = sin(r5569235);
        double r5569240 = r5569238 - r5569239;
        return r5569240;
}
↓
double f(double x, double eps) {
        double r5569241 = eps;
        double r5569242 = -9.918151501731852e-09;
        bool r5569243 = r5569241 <= r5569242;
        double r5569244 = x;
        double r5569245 = sin(r5569244);
        double r5569246 = cos(r5569241);
        double r5569247 = r5569245 * r5569246;
        double r5569248 = cos(r5569244);
        double r5569249 = sin(r5569241);
        double r5569250 = r5569248 * r5569249;
        double r5569251 = r5569247 + r5569250;
        double r5569252 = r5569251 - r5569245;
        double r5569253 = 3.630608855951668e-18;
        bool r5569254 = r5569241 <= r5569253;
        double r5569255 = 2.0;
        double r5569256 = r5569244 + r5569241;
        double r5569257 = r5569256 + r5569244;
        double r5569258 = r5569257 / r5569255;
        double r5569259 = cos(r5569258);
        double r5569260 = 0.5;
        double r5569261 = r5569241 * r5569260;
        double r5569262 = sin(r5569261);
        double r5569263 = r5569259 * r5569262;
        double r5569264 = r5569255 * r5569263;
        double r5569265 = r5569250 - r5569245;
        double r5569266 = r5569265 + r5569247;
        double r5569267 = r5569254 ? r5569264 : r5569266;
        double r5569268 = r5569243 ? r5569252 : r5569267;
        return r5569268;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.0
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -9.918151501731852e-09

    1. Initial program 29.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -9.918151501731852e-09 < eps < 3.630608855951668e-18

    1. Initial program 45.0

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

    3. Applied diff-sin45.0

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

    4. Simplified0.3

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

    if 3.630608855951668e-18 < eps

    1. Initial program 30.2

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

    3. Applied sin-sum1.2

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

    4. Applied associate--l+1.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.918151501731851694268208750014659269567 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 3.630608855951668331104028504087618457981 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))