Average Error: 37.1 → 0.5
Time: 22.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0771685445320532 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 1.3684517598693572 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.0771685445320532 \cdot 10^{-8}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 1.3684517598693572 \cdot 10^{-8}:\\
\;\;\;\;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 r131934 = x;
        double r131935 = eps;
        double r131936 = r131934 + r131935;
        double r131937 = sin(r131936);
        double r131938 = sin(r131934);
        double r131939 = r131937 - r131938;
        return r131939;
}
↓
double f(double x, double eps) {
        double r131940 = eps;
        double r131941 = -1.0771685445320532e-08;
        bool r131942 = r131940 <= r131941;
        double r131943 = x;
        double r131944 = sin(r131943);
        double r131945 = cos(r131940);
        double r131946 = r131944 * r131945;
        double r131947 = cos(r131943);
        double r131948 = sin(r131940);
        double r131949 = r131947 * r131948;
        double r131950 = r131949 - r131944;
        double r131951 = r131946 + r131950;
        double r131952 = 1.3684517598693572e-08;
        bool r131953 = r131940 <= r131952;
        double r131954 = 2.0;
        double r131955 = r131940 / r131954;
        double r131956 = sin(r131955);
        double r131957 = r131943 + r131940;
        double r131958 = r131957 + r131943;
        double r131959 = r131958 / r131954;
        double r131960 = cos(r131959);
        double r131961 = expm1(r131960);
        double r131962 = log1p(r131961);
        double r131963 = r131956 * r131962;
        double r131964 = r131954 * r131963;
        double r131965 = r131946 + r131949;
        double r131966 = r131965 - r131944;
        double r131967 = r131953 ? r131964 : r131966;
        double r131968 = r131942 ? r131951 : r131967;
        return r131968;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.2
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.0771685445320532e-08

    1. Initial program 30.4

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    if -1.0771685445320532e-08 < eps < 1.3684517598693572e-08

    1. Initial program 44.5

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

    3. Applied diff-sin44.5

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

    4. Simplified0.4

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

    6. Applied log1p-expm1-u0.5

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

    if 1.3684517598693572e-08 < eps

    1. Initial program 29.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0771685445320532 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 1.3684517598693572 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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)))