Average Error: 36.9 → 0.4
Time: 17.6s
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 r106795 = x;
        double r106796 = eps;
        double r106797 = r106795 + r106796;
        double r106798 = sin(r106797);
        double r106799 = sin(r106795);
        double r106800 = r106798 - r106799;
        return r106800;
}
↓
double f(double x, double eps) {
        double r106801 = eps;
        double r106802 = -8.79028981561156e-09;
        bool r106803 = r106801 <= r106802;
        double r106804 = x;
        double r106805 = sin(r106804);
        double r106806 = cos(r106801);
        double r106807 = r106805 * r106806;
        double r106808 = cos(r106804);
        double r106809 = sin(r106801);
        double r106810 = r106808 * r106809;
        double r106811 = r106807 + r106810;
        double r106812 = r106811 - r106805;
        double r106813 = 8.245159560248623e-09;
        bool r106814 = r106801 <= r106813;
        double r106815 = 2.0;
        double r106816 = fma(r106815, r106804, r106801);
        double r106817 = r106816 / r106815;
        double r106818 = cos(r106817);
        double r106819 = log1p(r106818);
        double r106820 = expm1(r106819);
        double r106821 = r106801 / r106815;
        double r106822 = sin(r106821);
        double r106823 = r106820 * r106822;
        double r106824 = r106823 * r106815;
        double r106825 = -r106805;
        double r106826 = fma(r106808, r106809, r106825);
        double r106827 = r106826 + r106807;
        double r106828 = r106814 ? r106824 : r106827;
        double r106829 = r106803 ? r106812 : r106828;
        return r106829;
}

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)))