Average Error: 37.0 → 0.5
Time: 12.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\
\;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2\\

\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 r116101 = x;
        double r116102 = eps;
        double r116103 = r116101 + r116102;
        double r116104 = sin(r116103);
        double r116105 = sin(r116101);
        double r116106 = r116104 - r116105;
        return r116106;
}
↓
double f(double x, double eps) {
        double r116107 = eps;
        double r116108 = -6.6823151206119754e-09;
        bool r116109 = r116107 <= r116108;
        double r116110 = x;
        double r116111 = sin(r116110);
        double r116112 = cos(r116107);
        double r116113 = r116111 * r116112;
        double r116114 = cos(r116110);
        double r116115 = sin(r116107);
        double r116116 = r116114 * r116115;
        double r116117 = r116116 - r116111;
        double r116118 = r116113 + r116117;
        double r116119 = 7.436477518387171e-09;
        bool r116120 = r116107 <= r116119;
        double r116121 = 2.0;
        double r116122 = r116107 / r116121;
        double r116123 = sin(r116122);
        double r116124 = r116110 + r116107;
        double r116125 = r116124 + r116110;
        double r116126 = r116125 / r116121;
        double r116127 = cos(r116126);
        double r116128 = r116123 * r116127;
        double r116129 = r116128 * r116121;
        double r116130 = r116113 + r116116;
        double r116131 = r116130 - r116111;
        double r116132 = r116120 ? r116129 : r116131;
        double r116133 = r116109 ? r116118 : r116132;
        return r116133;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.1
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 < -6.6823151206119754e-09

    1. Initial program 30.1

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -6.6823151206119754e-09 < eps < 7.436477518387171e-09

    1. Initial program 45.0

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

    3. Applied diff-sin45.0

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

    4. Simplified0.3

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

    if 7.436477518387171e-09 < eps

    1. Initial program 29.2

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))