Average Error: 37.2 → 0.5
Time: 14.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.106674806561576609834106951074650382338 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 1.744480835824209643740453456843553237121 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.106674806561576609834106951074650382338 \cdot 10^{-8}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 1.744480835824209643740453456843553237121 \cdot 10^{-8}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\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 r97120 = x;
        double r97121 = eps;
        double r97122 = r97120 + r97121;
        double r97123 = sin(r97122);
        double r97124 = sin(r97120);
        double r97125 = r97123 - r97124;
        return r97125;
}
↓
double f(double x, double eps) {
        double r97126 = eps;
        double r97127 = -1.1066748065615766e-08;
        bool r97128 = r97126 <= r97127;
        double r97129 = x;
        double r97130 = sin(r97129);
        double r97131 = cos(r97126);
        double r97132 = r97130 * r97131;
        double r97133 = cos(r97129);
        double r97134 = sin(r97126);
        double r97135 = r97133 * r97134;
        double r97136 = r97135 - r97130;
        double r97137 = r97132 + r97136;
        double r97138 = 1.7444808358242096e-08;
        bool r97139 = r97126 <= r97138;
        double r97140 = 2.0;
        double r97141 = r97126 / r97140;
        double r97142 = sin(r97141);
        double r97143 = r97129 + r97126;
        double r97144 = r97143 + r97129;
        double r97145 = r97144 / r97140;
        double r97146 = cos(r97145);
        double r97147 = r97142 * r97146;
        double r97148 = r97140 * r97147;
        double r97149 = r97132 + r97135;
        double r97150 = r97149 - r97130;
        double r97151 = r97139 ? r97148 : r97150;
        double r97152 = r97128 ? r97137 : r97151;
        return r97152;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
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 3 regimes

  2. if eps < -1.1066748065615766e-08

    1. Initial program 30.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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+0.7

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

    if -1.1066748065615766e-08 < eps < 1.7444808358242096e-08

    1. Initial program 44.6

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

    3. Applied diff-sin44.7

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

    4. Simplified0.3

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

    if 1.7444808358242096e-08 < eps

    1. Initial program 30.2

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.106674806561576609834106951074650382338 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 1.744480835824209643740453456843553237121 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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)))