Average Error: 37.0 → 0.4
Time: 14.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.179827039913242586237365435754276266067 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 4.118793419307951991804986633321323738488 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.179827039913242586237365435754276266067 \cdot 10^{-9}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 4.118793419307951991804986633321323738488 \cdot 10^{-10}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\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 r83245 = x;
        double r83246 = eps;
        double r83247 = r83245 + r83246;
        double r83248 = sin(r83247);
        double r83249 = sin(r83245);
        double r83250 = r83248 - r83249;
        return r83250;
}
↓
double f(double x, double eps) {
        double r83251 = eps;
        double r83252 = -9.179827039913243e-09;
        bool r83253 = r83251 <= r83252;
        double r83254 = x;
        double r83255 = sin(r83254);
        double r83256 = cos(r83251);
        double r83257 = r83255 * r83256;
        double r83258 = cos(r83254);
        double r83259 = sin(r83251);
        double r83260 = r83258 * r83259;
        double r83261 = r83260 - r83255;
        double r83262 = r83257 + r83261;
        double r83263 = 4.118793419307952e-10;
        bool r83264 = r83251 <= r83263;
        double r83265 = 2.0;
        double r83266 = r83251 / r83265;
        double r83267 = sin(r83266);
        double r83268 = r83254 + r83251;
        double r83269 = r83268 + r83254;
        double r83270 = r83269 / r83265;
        double r83271 = cos(r83270);
        double r83272 = expm1(r83271);
        double r83273 = log1p(r83272);
        double r83274 = r83267 * r83273;
        double r83275 = r83265 * r83274;
        double r83276 = r83257 + r83260;
        double r83277 = r83276 - r83255;
        double r83278 = r83264 ? r83275 : r83277;
        double r83279 = r83253 ? r83262 : r83278;
        return r83279;
}

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
Target14.5
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -9.179827039913243e-09

    1. Initial program 28.6

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.5

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

    if -9.179827039913243e-09 < eps < 4.118793419307952e-10

    1. Initial program 45.4

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

    3. Applied diff-sin45.4

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

    4. Simplified0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied log1p-expm1-u0.3

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}\right)\]

    if 4.118793419307952e-10 < eps

    1. Initial program 29.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.179827039913242586237365435754276266067 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 4.118793419307951991804986633321323738488 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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