Average Error: 37.1 → 0.5
Time: 24.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.496762876577124763011742607579646069382 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 9.496762876577124763011742607579646069382 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r59094 = x;
        double r59095 = eps;
        double r59096 = r59094 + r59095;
        double r59097 = sin(r59096);
        double r59098 = sin(r59094);
        double r59099 = r59097 - r59098;
        return r59099;
}
↓
double f(double x, double eps) {
        double r59100 = eps;
        double r59101 = -9.642672794131258e-05;
        bool r59102 = r59100 <= r59101;
        double r59103 = x;
        double r59104 = sin(r59103);
        double r59105 = cos(r59100);
        double r59106 = r59104 * r59105;
        double r59107 = cos(r59103);
        double r59108 = sin(r59100);
        double r59109 = r59107 * r59108;
        double r59110 = r59106 + r59109;
        double r59111 = r59110 - r59104;
        double r59112 = 9.496762876577125e-09;
        bool r59113 = r59100 <= r59112;
        double r59114 = 2.0;
        double r59115 = r59100 / r59114;
        double r59116 = sin(r59115);
        double r59117 = fma(r59114, r59103, r59100);
        double r59118 = r59117 / r59114;
        double r59119 = cos(r59118);
        double r59120 = r59116 * r59119;
        double r59121 = r59114 * r59120;
        double r59122 = r59109 - r59104;
        double r59123 = r59106 + r59122;
        double r59124 = r59113 ? r59121 : r59123;
        double r59125 = r59102 ? r59111 : r59124;
        return r59125;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.0
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 < -9.642672794131258e-05

    1. Initial program 30.2

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

    3. Applied sin-sum0.4

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

    if -9.642672794131258e-05 < eps < 9.496762876577125e-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. Simplified0.4

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

    if 9.496762876577125e-09 < eps

    1. Initial program 28.9

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

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.496762876577124763011742607579646069382 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))