Average Error: 37.0 → 0.5
Time: 22.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.4662243898819633 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 2.4662243898819633 \cdot 10^{-08}:\\
\;\;\;\;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 r8204110 = x;
        double r8204111 = eps;
        double r8204112 = r8204110 + r8204111;
        double r8204113 = sin(r8204112);
        double r8204114 = sin(r8204110);
        double r8204115 = r8204113 - r8204114;
        return r8204115;
}
↓
double f(double x, double eps) {
        double r8204116 = eps;
        double r8204117 = -2.082748610758504e-07;
        bool r8204118 = r8204116 <= r8204117;
        double r8204119 = x;
        double r8204120 = sin(r8204119);
        double r8204121 = cos(r8204116);
        double r8204122 = r8204120 * r8204121;
        double r8204123 = cos(r8204119);
        double r8204124 = sin(r8204116);
        double r8204125 = r8204123 * r8204124;
        double r8204126 = r8204122 + r8204125;
        double r8204127 = r8204126 - r8204120;
        double r8204128 = 2.4662243898819633e-08;
        bool r8204129 = r8204116 <= r8204128;
        double r8204130 = 2.0;
        double r8204131 = r8204116 / r8204130;
        double r8204132 = sin(r8204131);
        double r8204133 = r8204119 + r8204116;
        double r8204134 = r8204133 + r8204119;
        double r8204135 = r8204134 / r8204130;
        double r8204136 = cos(r8204135);
        double r8204137 = r8204132 * r8204136;
        double r8204138 = r8204130 * r8204137;
        double r8204139 = r8204129 ? r8204138 : r8204127;
        double r8204140 = r8204118 ? r8204127 : r8204139;
        return r8204140;
}

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 2 regimes

  2. if eps < -2.082748610758504e-07 or 2.4662243898819633e-08 < eps

    1. Initial program 29.8

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

    3. Applied sin-sum0.6

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

    if -2.082748610758504e-07 < eps < 2.4662243898819633e-08

    1. Initial program 44.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.082748610758504 \cdot 10^{-07}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.4662243898819633 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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)))