Average Error: 37.0 → 0.5
Time: 17.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.3997675699750705 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.9470983410706072 \cdot 10^{-13}:\\ \;\;\;\;\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.3997675699750705 \cdot 10^{-09}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 1.9470983410706072 \cdot 10^{-13}:\\
\;\;\;\;\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r2095183 = x;
        double r2095184 = eps;
        double r2095185 = r2095183 + r2095184;
        double r2095186 = sin(r2095185);
        double r2095187 = sin(r2095183);
        double r2095188 = r2095186 - r2095187;
        return r2095188;
}
↓
double f(double x, double eps) {
        double r2095189 = eps;
        double r2095190 = -7.3997675699750705e-09;
        bool r2095191 = r2095189 <= r2095190;
        double r2095192 = x;
        double r2095193 = cos(r2095192);
        double r2095194 = sin(r2095189);
        double r2095195 = r2095193 * r2095194;
        double r2095196 = sin(r2095192);
        double r2095197 = r2095195 - r2095196;
        double r2095198 = cos(r2095189);
        double r2095199 = r2095196 * r2095198;
        double r2095200 = r2095197 + r2095199;
        double r2095201 = 1.9470983410706072e-13;
        bool r2095202 = r2095189 <= r2095201;
        double r2095203 = 2.0;
        double r2095204 = fma(r2095203, r2095192, r2095189);
        double r2095205 = r2095204 / r2095203;
        double r2095206 = cos(r2095205);
        double r2095207 = r2095189 / r2095203;
        double r2095208 = sin(r2095207);
        double r2095209 = r2095203 * r2095208;
        double r2095210 = r2095206 * r2095209;
        double r2095211 = r2095202 ? r2095210 : r2095200;
        double r2095212 = r2095191 ? r2095200 : r2095211;
        return r2095212;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.5
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 2 regimes

  2. if eps < -7.3997675699750705e-09 or 1.9470983410706072e-13 < eps

    1. Initial program 30.3

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

    3. Applied sin-sum0.7

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

    4. Applied associate--l+0.7

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

    if -7.3997675699750705e-09 < eps < 1.9470983410706072e-13

    1. Initial program 44.3

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

    3. Applied diff-sin44.3

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

    4. Simplified0.2

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

    6. Applied associate-*r*0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.3997675699750705 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.9470983410706072 \cdot 10^{-13}:\\ \;\;\;\;\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))