Average Error: 37.3 → 0.4
Time: 14.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.379308090176633415808603902149090392193 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 6.52085484414657072417395637610113001692 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.379308090176633415808603902149090392193 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 6.52085484414657072417395637610113001692 \cdot 10^{-9}\right):\\
\;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\

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

\end{array}
double f(double x, double eps) {
        double r118392 = x;
        double r118393 = eps;
        double r118394 = r118392 + r118393;
        double r118395 = sin(r118394);
        double r118396 = sin(r118392);
        double r118397 = r118395 - r118396;
        return r118397;
}
↓
double f(double x, double eps) {
        double r118398 = eps;
        double r118399 = -1.3793080901766334e-08;
        bool r118400 = r118398 <= r118399;
        double r118401 = 6.520854844146571e-09;
        bool r118402 = r118398 <= r118401;
        double r118403 = !r118402;
        bool r118404 = r118400 || r118403;
        double r118405 = x;
        double r118406 = cos(r118405);
        double r118407 = sin(r118398);
        double r118408 = sin(r118405);
        double r118409 = -r118408;
        double r118410 = fma(r118406, r118407, r118409);
        double r118411 = cos(r118398);
        double r118412 = r118408 * r118411;
        double r118413 = r118410 + r118412;
        double r118414 = 2.0;
        double r118415 = r118398 / r118414;
        double r118416 = sin(r118415);
        double r118417 = fma(r118414, r118405, r118398);
        double r118418 = r118417 / r118414;
        double r118419 = cos(r118418);
        double r118420 = r118416 * r118419;
        double r118421 = r118414 * r118420;
        double r118422 = r118404 ? r118413 : r118421;
        return r118422;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target14.9
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 2 regimes

  2. if eps < -1.3793080901766334e-08 or 6.520854844146571e-09 < eps

    1. Initial program 29.9

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

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

    5. Simplified0.5

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

    if -1.3793080901766334e-08 < eps < 6.520854844146571e-09

    1. Initial program 44.9

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

    3. Applied diff-sin44.9

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

    4. Simplified0.3

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

    6. Applied log1p-expm1-u0.4

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

    7. Simplified0.4

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\right)\right)\]
    8. Using strategy rm

    9. Applied log1p-expm10.3

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \color{blue}{\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.379308090176633415808603902149090392193 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 6.52085484414657072417395637610113001692 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(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)))