Average Error: 36.9 → 0.4
Time: 15.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.790289815611559393126472092098999677745 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.245159560248622983676760917360476499383 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.790289815611559393126472092098999677745 \cdot 10^{-9}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 8.245159560248622983676760917360476499383 \cdot 10^{-9}:\\
\;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\

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

\end{array}
double f(double x, double eps) {
        double r84705 = x;
        double r84706 = eps;
        double r84707 = r84705 + r84706;
        double r84708 = sin(r84707);
        double r84709 = sin(r84705);
        double r84710 = r84708 - r84709;
        return r84710;
}
↓
double f(double x, double eps) {
        double r84711 = eps;
        double r84712 = -8.79028981561156e-09;
        bool r84713 = r84711 <= r84712;
        double r84714 = x;
        double r84715 = sin(r84714);
        double r84716 = cos(r84711);
        double r84717 = r84715 * r84716;
        double r84718 = cos(r84714);
        double r84719 = sin(r84711);
        double r84720 = r84718 * r84719;
        double r84721 = r84717 + r84720;
        double r84722 = r84721 - r84715;
        double r84723 = 8.245159560248623e-09;
        bool r84724 = r84711 <= r84723;
        double r84725 = 2.0;
        double r84726 = fma(r84725, r84714, r84711);
        double r84727 = r84726 / r84725;
        double r84728 = cos(r84727);
        double r84729 = log1p(r84728);
        double r84730 = expm1(r84729);
        double r84731 = r84711 / r84725;
        double r84732 = sin(r84731);
        double r84733 = r84730 * r84732;
        double r84734 = r84733 * r84725;
        double r84735 = -r84715;
        double r84736 = fma(r84718, r84719, r84735);
        double r84737 = r84736 + r84717;
        double r84738 = r84724 ? r84734 : r84737;
        double r84739 = r84713 ? r84722 : r84738;
        return r84739;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.2
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 < -8.79028981561156e-09

    1. Initial program 29.5

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -8.79028981561156e-09 < eps < 8.245159560248623e-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 expm1-log1p-u0.3

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

    7. Simplified0.3

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

    if 8.245159560248623e-09 < eps

    1. Initial program 29.7

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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.790289815611559393126472092098999677745 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.245159560248622983676760917360476499383 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \sin \varepsilon, -\sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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)))