Average Error: 36.1 → 0.4
Time: 21.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.6103683390120787 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 4.3324917171030357 \cdot 10^{-07}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.6103683390120787 \cdot 10^{-05}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 4.3324917171030357 \cdot 10^{-07}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{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 r4405135 = x;
        double r4405136 = eps;
        double r4405137 = r4405135 + r4405136;
        double r4405138 = sin(r4405137);
        double r4405139 = sin(r4405135);
        double r4405140 = r4405138 - r4405139;
        return r4405140;
}
↓
double f(double x, double eps) {
        double r4405141 = eps;
        double r4405142 = -5.6103683390120787e-05;
        bool r4405143 = r4405141 <= r4405142;
        double r4405144 = x;
        double r4405145 = sin(r4405144);
        double r4405146 = cos(r4405141);
        double r4405147 = r4405145 * r4405146;
        double r4405148 = cos(r4405144);
        double r4405149 = sin(r4405141);
        double r4405150 = r4405148 * r4405149;
        double r4405151 = r4405147 + r4405150;
        double r4405152 = r4405151 - r4405145;
        double r4405153 = 4.3324917171030357e-07;
        bool r4405154 = r4405141 <= r4405153;
        double r4405155 = 2.0;
        double r4405156 = 0.5;
        double r4405157 = r4405156 * r4405141;
        double r4405158 = sin(r4405157);
        double r4405159 = r4405144 + r4405141;
        double r4405160 = r4405159 + r4405144;
        double r4405161 = r4405160 / r4405155;
        double r4405162 = cos(r4405161);
        double r4405163 = r4405158 * r4405162;
        double r4405164 = r4405155 * r4405163;
        double r4405165 = r4405150 - r4405145;
        double r4405166 = r4405165 + r4405147;
        double r4405167 = r4405154 ? r4405164 : r4405166;
        double r4405168 = r4405143 ? r4405152 : r4405167;
        return r4405168;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.1
Target14.2
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 < -5.6103683390120787e-05

    1. Initial program 29.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -5.6103683390120787e-05 < eps < 4.3324917171030357e-07

    1. Initial program 43.6

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

    3. Applied diff-sin43.6

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

    4. Simplified0.4

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

    if 4.3324917171030357e-07 < eps

    1. Initial program 28.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.6103683390120787 \cdot 10^{-05}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 4.3324917171030357 \cdot 10^{-07}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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